Complex logarithms. Significance and power Complex numbers logarithmic function for teapots

Appointment of that power

Complex zero does not have a logarithm, complex exponential scales do not take on a zero value. Nenulove texvc can be presented in the display form:

It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - math/README - z=r \cdot e^(i (\varphi + 2 \pi k))\;\;, de It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - finalizing the update.): k- quite a number

Todi It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - math/readme: \mathrm(Ln)\,z know the formula:

It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - math/readme: \mathrm(Ln)\,z = \ln r + i \left(\varphi + 2 \pi k \right)

Here It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - completion of the setup.): \ln\,r=\ln\,|z|- speech logarithm. Sounds are screaming:

From the formula it is clear that in one and less than one z the value is clearly visible part of the change in the interval It is impossible to open the virus (winning file texvc . The meaning is called leading values complex natural logarithm. Vidpovidna (already unambiguous) function is called head gulka the logarithm is indicated It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - math/README: \ln\,z. Inode through It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - math/README - tweaking.): \ln\, z also indicate the value of the logarithm, which lies not so on the head of the head. Yakscho It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - completion of the upgrade.): z- a speech number, then the smut value of the yogo logarithm is taken with the greatest speech logarithm.

From the induced formula, it is also clear that the speech part of the logarithm is marked by an offensive rank through the component of the argument:

It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - math/README: \operatorname(Re)(\ln(x+iy)) = \frac(1)(2) \ln(x^2+y^2)

The small picture shows that the speech part, as a function of the components, is centrally symmetrical and lies only a little way up to the cob of coordinates. The wrapper of the graph of the speech logarithm will appear on the vertical axis. Close to zero, the function of the right It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - finalizing the update.): -\infty.

The logarithm of a negative number is behind the formula:

It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - math/readme: \mathrm(Ln) (-x) = \ln x + i \pi (2 k + 1) \qquad (x>0,\ k = 0, \pm 1 , \ pm2\dots)

Apply the value of the complex logarithm

Introducing the smut value of the logarithm ( It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - finalization.): \ln) that fierce yoga expression ( It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - math/readme: \mathrm(Ln)) for valid arguments:

It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - tweaking.): \ln (1) = 0;\; \mathrm(Ln) (1) = 2k\pi i It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - finalization.): \ln (-1) = i \pi;\; \mathrm(Ln) (-1) = (2k+1)i \pi It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - math/README: \ln (i) = i \frac(\pi) (2);\; \mathrm(Ln) (i) = i \frac(4k+1)(2) \pi

Let's keep it safe when converting complex logarithms, vrakhovuyuchi, because the smells are rich, and therefore, the equality of the logarithms of any virus does not show the equality of these viruses. Butt pardon mirkuvannya:

It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - finalization: i\pi = \ln(-1) = \ln((-i)^2) = 2\ln(-i) = 2(-i\pi/2 ) = -i\pi- Obviously a pardon.

It is significant that the left-hander is worth the head value of the logarithm, and the right-hander is the value of the lower head ( It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - finalization: k=-1). The reason for the pardon is the carelessness of the authorities It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - math/README - tweaking.): \log_a((b^p)) = p~\log_a b.

Complex logarithmic function and Riemann surface

By virtue of the single linkage, the Riemann surface of the logarithm is a universal curve for the complex plane without a speck. It is impossible to open the virus (winning file texvc .

Analytical follow-up

The logarithm of a complex number can also be assigned as an analytical extension of the speech logarithm to the entire complex plane. Let's go crooked It is impossible to open the virus (winning file texvc start in loneliness, do not pass through zero and do not change the negative part of the speech axis. Same smut value of the logarithm in final points It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - finalization of the adjustment.): w curve It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - math/README: \Gamma can be assigned to the formula:

It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - tweaking.): \ln z = \int\limits_\Gamma (du \over u)

Yakscho It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - math/README: \Gamma- a simple curve (without self-permutation), then for numbers that lie on it, the logarithmic totemness can be fixed without a fight, for example:

It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - finalization: \ln (wz) = \ln w + \ln z, ~\forall z,w\in\Gamma\colon zw\in \Gamma

The head of the logarithmic function is uninterrupted and differentiated on the entire complex plane, but on the negative part of the speech axis, on the It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - finalizing the update.): 2\pi. And this fact is the result of a piece-wise exchange of the obvious part of the head value with an interval It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - finalizing the setup.): (-\pi, \pi]. If you look at all the halls of the function, then the continuity can be at all points, except for zero, the de function is not assigned. How to allow a crooked It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - math/README: \Gamma to shift the negative part of the speech axis, then first to transfer the result from the head value to the susceptus of the head, and on the skin step, the result is similar to the displacement of the logarithmic function along the legs (div. babies).

3 formulas for analytical follow-up, what should be the logarithm's logarithm:

It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - tweaking.): \frac(d)(dz) \ln z = (1\over z)

For whatever stake It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - finalization of the finalization.): S, what is the point It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - finalizing the update.): 0 :

It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - tweaking.): \oint\limits_S (dz \over z) = 2\pi i

The integral is taken from the positive direction (anti-Year arrow). The value lies at the basis of the theory of recovery.

You can also calculate the analytical continuation of the complex logarithm for additional rows, used for the speech pattern:

Prote z mind tsikh ryadіv viplivaє, scho in unity sum to a number of dorivnyuє zero, so the series is brought only up to the head of the rich-valued function of the complex logarithm. Radius of living on both rows of doors 1.

Links with inverted trigonometric and hyperbolic functions

It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - completion of the setup.): \operatorname(Arcsin) z = -i \operatorname(Ln) (і z + \sqrt(1-z^2)) It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - math/README - tweaking.): \operatorname(Arccos) z = -i \operatorname(Ln) (z + i\sqrt(1-z^2)) It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - finalization.): \operatorname(Arctg) z = -\frac(i)(2) \ln \frac(1+z i)(1-z i) + k \pi \; (z \ne \pm i) It is impossible to open the virus (winning file texvc NOT knowledge; math/README - math/README - tweaking.): \operatorname(Arcctg) z = -\frac(i)(2) \ln \frac(z i-1)(z i+1) + k \pi \; (z \ne \pm i) It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - completion of the setup.): \operatorname(Arsh)z = \operatorname(Ln)(z+\sqrt(z^2+1))- reverse hyperbolic sine It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - completion of the setup.): \operatorname(Arch)z=\operatorname(Ln) \left(z+\sqrt(z^(2)-1) \right)- reverse hyperbolic cosine It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - finalizing the setup.): \operatorname(Arth)z=\frac(1)(2)\operatorname(Ln)\left(\frac(1+z)(1-z)\right)- reverse hyperbolic tangent It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - completion of the setup.): \operatorname(Arcth)z=\frac(1)(2)\operatorname(Ln)\left(\frac(z+1)(z-1)\right)- inverted hyperbolic cotangent

historical drawing

The first to try to solve logarithms for complex numbers at the turn of the XVII-XVIII century, Leibniz and Johann Bernoulli, however, did not succeed in creating a coherent theory - for these reasons, that even then it was not clear to understand the logarithm itself. The discussion on this subject was carried on together between Leibniz and Bernoulli, and in the middle of the 18th century between d'Alembert and Euler. Bernoulli and d'Alembert considered what should be It is impossible to open the virus (winning file texvc NOT knowledge; Div. math/README - tweaking.): \log(-x) = \log(x), similarly to Leibnitz, having proved that the logarithm of a negative number is a clear number . The new theory of the logarithms of negative and complex numbers was published by Euler in 1747-1751 and, in fact, is not considered in any way today. Wanting a super girl trival (D'Alembert, having defended his thought and reportedly arguing them in the articles of his "Encyclopedia" and other practices), Euler's pidkhid until the end of the 18th century took away the deep knowledge.

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Literature

Theory of logarithms

• Korn R., Korn T.. – M.: Nauka, 1973. – 720 p.
• Sveshnikov A. G., Tikhonov A. M. Theory of functions of complex change. – M.: Nauka, 1967. – 304 p.
• Fikhtengolts G.M. Course of differential and integral calculation. - View. 6th. – M.: Nauka, 1966. – 680 p.

History of logarithms

• Mathematics of the 18th century // Under the editorship of A.P. Yushkevich, in three volumes. - M.: Nauka, 1972. - T. III.
• Kolmogorov A. N., Yushkevich A. P. (ed.). Mathematics of the XIX century. Geometry. Theory of analytic functions. - M.: Nauka, 1981. - T. II.

Notes

1. Logarithmic function. //. - M .: Radianska Encyclopedia, 1982. - T. 3.
2. , Volume II, pages 520-522.
3. , with. 623..
4. , with. 92-94.
5. , with. 45-46, 99-100.
6. Boltyansky St R., Efremovich St.. - M.: Nauka, 1982. - P. 112. - (Quantum Library, issue 21).
7. , Volume II, pages 522-526.
8. , with. 624..
9. , with. 325-328.
10. Ribnikov K. A. History of mathematics. In two volumes. - M .: View. MDU, 1963. - T. II. - S. 27, 230-231.
11. , with. 122-123.
12. Klein F.. - M.: Nauka, 1987. - T. II. Geometry. - S. 159-161. – 416 p.

Lesson that characterizes the complex logarithm

Seeing the wild heat that suffocated us, they rushed in sacks through a wide valley, not thinking about those who could drink a little on another “above” ... We simply didn’t have an hour to think about it - we already shackled.
The creature flew right above us, loudly clattering with its toothy jaw, and we rushed, rushing out of strength, spraying disgusting slime drizzles in the direction, and good thoughts, so that the engine of the "wonder bird" would chirp even more quickly... richer swedish and spying in it, we simply didn’t have any chances. As for evil, not a single tree grew nearby, there were no bushes, no stones, for which it would be possible to hide, only in the distance one could see the dashing black of the skelya.
- Tudy! - Pointing a finger at the same skeleton, Stella screamed.
Ale raptom, unsurprisingly, right in front of us, the sound of stota appeared, in the form of which it literally caught our blood in our veins ... It vinykla nibi "right out of the air" and was in a right shriveled way ... The majestic black carcass zhorstko hair, rolyachi yogo similar to a pot-bellied bear, only this “wedmid” was as tall as a three-surfaced booty ... The bumpy head of the monster “crumpled” with two majestic curved horns, and a pair of nim- ovirously long, gostrikh, like knives, іkol, marveled at on the yak, the legs swayed like a sling... And here, invisibly raising us, the monster easily streaked up the hill and .... having lifted the flying "gidota" to one of its majestic cleats... We were stunned.
- Bіzhimo! Stella squealed. - Tіkaєmo, until the wines of "occupations"!
And we were already ready to go again without looking back, like a raptom piercing a thin voice behind our backs:
- Girl, check it out! Don't need to tick!
They turned around sharply - a crying girl stood behind her, a beautiful black-eyed girl ... and calmly sang miraculously, what had come to her! Well, it’s definitely a surprise day! .. The girl, marveling at us, smiled warmly, we’re not afraid of the order of the hairy monster, who should be instructed.
- Be kind, do not fight yoga. Vin is kind. We yelled that Ovara was chasing you, and they helped us. Dean the young man caught up at once. Really, my dear?
“Garniy” grumbled, pierced like a light earthworm, and shaved his head, licking the girl in disguise.
- And who is Ovara, and why did she attack us? - I asked.
- Vaughn attacks everyone, she is a hut. I’m not safe anymore,” the girl said calmly. - Can you sleep, what are you doing here? Don't you see, girls?
- No, don't look. We were just walking. Ale, well, food is up to you - and why are you robish here?
I’m going to mami ... - the little one frowned. - We died at once, and yet, I think, she drank suddi. And now I live here, but I don’t say anything, because there’s no time for it. You think that I'm just coming...
- Why don't you come more often and rightly? It’s so greedy right there! .. - Stella exchanged her pleats.
– I can’t leave one here, I wonder after her that nothing has become of her. І axis Dіn zі me... Vіn help me.
I just couldn’t believe it... This little good girl voluntarily went out of her warm and kind “above”, to live in this cold, stingy and alien world, protecting her own, supposedly “vinnut”, mother! It’s not rich, I think, I would have known such good and self-confident (to grow up!) people who would have been called to such a feat ... I immediately thought - maybe, she just didn’t understand what she was going to say to herself ?!
- And how long have you been here, girl, is it not a secret?
- Recently ... - summarily vіdpovila, smacking with her fingers the black curl of her curly hair, black-eyed little. - I drank in such a warm world, if I died! .. Vin was so kind and light! I was so scared! Її why didn’t it happen anywhere... And then I fell into this zhahlivy world... And then її knew. I was so scared here ... So naturally ... Mom told me to go, she barked. Ale, I can't get rid of it ... Now I have a friend, my kind Dean, and I can already feel like I'm here.
The “good friend” became livid again, as Stella gave us majestic “lower astral” goosebumps... Having got up, I tried to calm down a little, and began to marvel at that hairy diva... they turned back their respect, motorically baring their teeth at their pasture... I jumped out.
- Oh, don't fight, be kind! Why should you smile, - the girl calmed down.
So... With such a laugh, you'll start to learn a lot... - I thought.
- And how did it happen, what did you talk to him about? - Asked Stella.
- If I only came here, I was even more scared, especially if such monsters attacked you today. And once, if I didn’t die a little, Dean, having turned me in like a whole, buy terrible flying “birds”. I also snarled at the back of my head, but then I understood, like a new golden heart ... Vin is the best friend! I didn’t have such ones, if I lingered on the Earth.
- But how did you sound so brightly before the new one? Aje, the new one’s callousness is not familiar, let’s put it this way, it’s zvichna ...
- And here I learned one simple truth, I didn’t remember what I thought on the Earth - calliness doesn’t matter, because people have a good heart... And then all the beauty disappeared somewhere ... And Din, even though it’s terrible, then forever be kind, and forever protect me, I see his goodness and I’m not afraid of anything. And you can call to zovnіshnosti.
- Do you know that you will be here for a long time, more richly, than people live on earth? Why don't you want to stay here?
- Here is my mother, father, I'm mushu help him. And if it’s “going out”, so that I can live again on Earth, I’ll also go ... There, for goodness, more. In this terrible world, people are even more wondrous - they don’t stink and don’t linger in fire. Why so? Do you know about it?
- And who told you that your mother is gone, so that I can live again? - Stella got stuck.
- Dean, obviously. You know a lot, even if you've been living here for a long time. And having said that if we (my mother and I) live again, we will have other people with us. And then in me you won’t be mothers anymore ... Then I want to hang out with her at once.
- And how do you speak to him, to your Dean? - Asked Stella. – Why don’t you want to say your name?
And it's true - we didn't know how it sounded! And the stars didn't even know...
- My name was Maria ... But what is the meaning of this?
- Well, obviously! - Stella laughed. - And how can I talk to you? Axis if you come - there you will name a new one, but for now, stay here, you will live in old age. Did you speak here from Kimos, girl Marie? - For the zvichkoyu jumping from those to the topic, Stella asked.
- So, I was talking ... - the girl said indifferently. - Ale, the stench here is so marvellous. And so unfortunate... Why stink so unhappy?
- And hіba those, what are you doing here, can you be happy? - I was happy to feed. - Bring on the local “reality” itself, driving in the future be-like hopes! .. How can you be happy here?
- I do not know. If I’m with my mother, I’m happy, I could be happy here too ... It’s true, it’s scary here, and I don’t fit here anymore ... If I said that I’m good for her, she yelled at me a lot and said , that I am "unfortunately brainless" ... But I don’t show myself ... I know that I’m just scared. So, just like me...
- Maybe, she just wanted to save you from your “extreme” decision, and wanted you to go back to her “above”? - Carefully, schob not ugly, Stella was sleeping.
- No, zvichayno well ... Ale, thank you for your kind words. Mom often called me not zovsі good names, navіt on the Earth ... But I know that it's not zі malice. Vaughn was simply unfortunate through those that I was born, and often told me that I ruined my life. But it wasn't my fault, was it? I always tried to be happy, but why didn’t I even go into it ... But there was no such thing as a tata in me. - Maria was more sumptuous, and her voice was tremtiv, instead of the axle-axis of crying.
Mi and Stella looked at each other, and I was even more impressed, that I saw similar thoughts... on top of that, even more painfully false.
- Axis Dіn seems that I'm good, and that I'm working yogo even more happy! - The little one stammered more cheerfully. - I want to be friends with me. And others, whom I've taunted here, are even colder than baiduzhi, and sometimes they bring evil ... Especially those who have monsters attached ...
- Monsters - what? .. - we did not understand.
- Well, they have scary monsters on their backs and seem to them that the stench can be robit. And if you don’t listen, the monsters are terribly worried about them ... I tried to talk to them, but the monsters won’t allow it.
We didn’t understand absolutely anything of this “explanation”, but the very fact that such astral things are shaking people, we didn’t “accomplish” in a moment, that’s why we were fed, as we may be, a marvelous manifestation of magic.
- Oh, that creak! Especially bіlya "black burn". There are wines, behind the trees. Wait, mi tezh with you pіdemo?
- Zvichayno, for my sake! - Stella was immediately greeted.
Meni tezh, to be honest, did not even laugh at the prospect of zustrichatsya with a kimoshche, “motor-powered and unreasonable”, especially self-taught. Aleksey tsіkavіst overcame fear, and we, obviously, would have gone, regardless of those who were afraid of the little ones ... But if we had such a zahisnik like Din with us, it would have been more fun ...
First axis, in a short space of time, in front of our wide-opened eyes, right Hell flared up... ... Divine wines, obviously, not buv, but bov just to succumb, as if I could succumb to the lower Astral. Ale, you need to watch out for you more - having depicted the wine of yoga miraculously ... I scribbled yoga pictures in the book, as a bule in the library of my father, and remembered those motors in my memory, as if they carried in themselves more of yoga paintings ...
- What a blast! .. - Stella whispered in her nakedness.
It would be possible, singly, it would be possible to say that we were running here, on the “tops”, it’s already rich ... But it’s not possible for us to see such a thing in our worst nightmare! It looked like a majestic, waving near the rock, a flat “cauldron”, on the bottom of which crimson “lava” bubbled up ... The rozpechenie “burst” everywhere with marvelous red puffs, which they spalahuly, from which the singing steam was surviving and fell with great drops. on the ground, or on people, as if at that moment they lifted themselves up ... Unselfish cries lulled, but they immediately moved, more on the backs of quiet people sat ogid creatures, as if with a satisfied look, they “cheered” their victims, not brutally greedy on these sufferings... Under the bare feet of people, a red stone was baked, bubbled and "melted" food with the heat of the crimson earth... Crisis majestic cracks erupted with splashes of hot steam and, scorching the feet of human beings, which were all over the place, pain. hanging, vaping with a light smoky... And along the very middle of the "pit" ran a bright-red, wide, fiery river, at yak, hour after hour, those ogidnі monsters unstoppably bubbling over that chi іnsha tormented day, yak, falling, called out more a short splash of orange sparks, and right there, disguised as a mistress in a fluffy white haze, she knew ... already for a while ... It was hellish, and we, Stella, wanted to know how to “get in touch” ...
– What is robitimem?.. – Stella whispered in a quiet feverish voice. - Do you want to go down there? Hiba can we help them? Marvel at how rich it is!
We stood on a black-brown, heat-dried shave, guarded by the pain, lack of escape, and violence, and felt the floorings were childishly powerless, which my warrior Stella this time categorically folded and was ready to bula at the first call rush to your own, so dear and above, the upper "on top" ...

Material from Wikipedia - free encyclopedia

Appointment of that power

Complex zero does not have a logarithm, complex exponential scales do not take on a zero value. Nenulove $z$ can be presented in the display form:

$z=r\; \backslash cdot\; e^(i\; (\backslash varphi\; +\; 2\; \backslash pi\; k))\backslash ;\backslash ;,$ de $k$- quite a number

Todi $\backslash mathrm(Ln)\backslash ,z$ know the formula:

$\backslash mathrm(Ln)\backslash ,z\; =\; \backslash ln\; r\; +\; i\; \backslash left(\backslash varphi\; +\; 2\; \backslash pi\; k\; \backslash right)$

Here $\backslash ln\backslash ,r=\; \backslash ln\backslash ,|z|$- speech logarithm. Sounds are screaming:

$\backslash mathrm(Ln)\; (-x)\; =\; \backslash ln\; x\; +\; i\; \backslash pi\; (2\; k\; +\; 1)\; \backslash qquad\; (x>0,\backslash \; k\; =\; 0,\; \backslash pm\; 1,\; \backslash pm\; 2\; \backslash dots)$

Apply the value of the complex logarithm

Introducing the smut value of the logarithm ( $\backslash ln$) that fierce yoga expression ( $\backslash mathrm(Ln)$) for valid arguments:

$\backslash ln(1)\; =\; 0;\backslash ;\; \backslash mathrm(Ln)\; (1)\; =\; 2k\backslash pi\; i$ $\backslash ln\; (-1)\; =\; i\; \backslash pi;\backslash ;\; \backslash mathrm(Ln)\; (-1)\; =\; (2k+1)i\; \backslash pi$ $\backslash ln(i)\; =\; i\; \backslash frac(\backslash pi)\; (2);\backslash ;\; \backslash mathrm(Ln)\; (i)\; =\; i\; \backslash frac(4k+1)(2)\; \backslash pi$

Let's keep it safe when converting complex logarithms, vrakhovuyuchi, because the smells are rich, and therefore, the equality of the logarithms of any virus does not show the equality of these viruses. Butt pardon mirkuvannya:

$i\backslash pi\; =\; \backslash ln(-1)\; =\; \backslash ln((-i)^2)\; =\; 2\backslash ln(-i)\; =\; 2(-i\backslash pi/2)\; =\; -i\backslash pi$- Obviously a pardon.

It is significant that the left-hander is worth the head value of the logarithm, and the right-hander is the value of the lower head ( $k=-1$). The reason for the pardon is the carelessness of the authorities $\backslash log\_a((b^p))\; =\; p~\backslash log\_a\; b$.

Complex logarithmic function and Riemann surface

By virtue of the single linkage, the Riemann surface of the logarithm is a universal curve for the complex plane without a speck. $0$.

Analytical follow-up

The logarithm of a complex number can also be assigned as an analytical extension of the speech logarithm to the entire complex plane. Let's go crooked $\backslash Gamma$ start in loneliness, do not pass through zero and do not change the negative part of the speech axis. Same smut value of the logarithm in final points $w$ curve $\backslash Gamma$ can be assigned to the formula:

$\backslash ln\; z\; =\; \backslash int\backslash limits\_\backslash Gamma\; (du\; \backslash over\; u)$

Yakscho $\backslash Gamma$- a simple curve (without self-permutation), then for numbers that lie on it, the logarithmic totemness can be fixed without a fight, for example:

$\backslash ln\; (wz)\; =\; \backslash ln\; w\; +\; \backslash ln\; z,\; ~\backslash forall\; z,w\backslash in\backslash Gamma\backslash colon\; zw\backslash in\; \backslash Gamma$

The head of the logarithmic function is uninterrupted and differentiated on the entire complex plane, but on the negative part of the speech axis, on the $2\backslash pi$. And this fact is the result of a piece-wise exchange of the obvious part of the head value with an interval $(-\backslash pi,\; \backslash pi]$. If you look at all the halls of the function, then the continuity can be at all points, except for zero, the de function is not assigned. How to allow a crooked $\backslash Gamma$ to shift the negative part of the speech axis, then first to transfer the result from the head value to the susceptus of the head, and on the skin step, the result is similar to the displacement of the logarithmic function along the legs (div. babies).

3 formulas for analytical follow-up, what should be the logarithm's logarithm:

$\backslash frac(d)(dz)\; \backslash ln\; z\; =\; (1\backslash over\; z)$

For whatever stake $S$, what is the point $0$:

$\backslash oint\backslash limits\_S\; (dz\; \backslash over\; z)\; =\; 2\backslash pi\; i$

The integral is taken from the positive direction (anti-Year arrow). The value lies at the basis of the theory of recovery.

You can also calculate the analytical continuation of the complex logarithm for additional rows, used for the speech pattern:

{{{2}}}

(Row 1)

{{{2}}}

(Row 2)

Prote z mind tsikh ryadіv viplivaє, scho in unity sum to a number of dorivnyuє zero, so the series is brought only up to the head of the rich-valued function of the complex logarithm. Radius of living on both rows of doors 1.

Links with inverted trigonometric and hyperbolic functions

$\backslash operatorname(Arcsin)\; z\; =\; -i\; \backslash operatorname(Ln)\; (і\; z\; +\; \backslash sqrt(1-z^2))$ $\backslash operatorname(Arccos)\; z\; =\; -i\; \backslash operatorname(Ln)\; (z\; +\; i\backslash sqrt(1-z^2))$ $\backslash operatorname(Arctg)\; z\; =\; -\backslash frac(i)(2)\; \backslash ln\; \backslash frac(1+z\; i)(1-z\; i)\; +\; k\; \backslash pi\; \backslash ;\; (z\; \backslash ne\; \backslash pm\; i)$ $\backslash operatorname(Arcctg)\; z\; =\; -\backslash frac(i)(2)\; \backslash ln\; \backslash frac(z\; i-1)(z\; i+1)\; +\; k\; \backslash pi\; \backslash ;\; (z\; \backslash ne\; \backslash pm\; i)$ $\backslash operatorname(Arsh)z\; =\; \backslash operatorname(Ln)(z+\backslash sqrt(z^2+1))$- reverse hyperbolic sine $\backslash operatorname(Arch)z=\backslash operatorname(Ln)\; \backslash left(z+\backslash sqrt(z^(2)-1)\; \backslash right)$- reverse hyperbolic cosine $\backslash operatorname(Arth)z=\backslash frac(1)(2)\backslash operatorname(Ln)\backslash left(\backslash frac(1+z)(1-z)\backslash right)$- reverse hyperbolic tangent $\backslash operatorname(Arcth)z=\backslash frac(1)(2)\backslash operatorname(Ln)\backslash left(\backslash frac(z+1)(z-1)\backslash right)$- inverted hyperbolic cotangent

historical drawing

The first to try to solve logarithms for complex numbers at the turn of the XVII-XVIII century, Leibniz and Johann Bernoulli, however, did not succeed in creating a coherent theory - for these reasons, that even then it was not clear to understand the logarithm itself. The discussion on this subject was carried on together between Leibniz and Bernoulli, and in the middle of the 18th century between d'Alembert and Euler. Bernoulli and d'Alembert considered what should be $\backslash log(x)\; =\; \backslash log(x)$, similarly to Leibnitz, having proved that the logarithm of a negative number is a clear number . The new theory of the logarithms of negative and complex numbers was published by Euler in 1747-1751 and, in fact, is not considered in any way today. Wanting a super girl trival (D'Alembert, having defended his thought and reportedly arguing them in the articles of his "Encyclopedia" and other practices), Euler's pidkhid until the end of the 18th century took away the deep knowledge.

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Literature

Theory of logarithms

• Korn R., Korn T.. – M.: Nauka, 1973. – 720 p.
• Sveshnikov A. G., Tikhonov A. M. Theory of functions of complex change. – M.: Nauka, 1967. – 304 p.
• Fikhtengolts G.M. Course of differential and integral calculation. - View. 6th. – M.: Nauka, 1966. – 680 p.

History of logarithms

• Mathematics of the 18th century // Under the editorship of A.P. Yushkevich, in three volumes. - M.: Nauka, 1972. - T. III.
• Kolmogorov A. N., Yushkevich A. P. (ed.). Mathematics of the XIX century. Geometry. Theory of analytic functions. - M.: Nauka, 1981. - T. II.

Notes

1. Logarithmic function. //. - M .: Radianska Encyclopedia, 1982. - T. 3.
2. , Volume II, pages 520-522.
3. , with. 623..
4. , with. 92-94.
5. , with. 45-46, 99-100.
6. Boltyansky St R., Efremovich St.. - M.: Nauka, 1982. - P. 112. - (Quantum Library, issue 21).
7. , Volume II, pages 522-526.
8. , with. 624..
9. , with. 325-328.
10. Ribnikov K. A. History of mathematics. In two volumes. - M .: View. MDU, 1963. - T. II. - S. 27, 230-231.
11. , with. 122-123.
12. Klein F.. - M.: Nauka, 1987. - T. II. Geometry. - S. 159-161. – 416 p.

Lesson that characterizes the complex logarithm

It can be seen that this strong, wondrous person, having changed under an enchanting infusion, turns on a new black, graceful, loving girl.
Rostov commemorated the new world of Dolokhov and Sonya; ale without signing yourself, yakі tse buli new vіdnosiny. “Stink there you mustache into a cohosh,” thinking wine about Sonya and Natasha. But it was not so for you, as before, in peace with Sonya and Dolokhov, and sooner becoming at home.
In the autumn of 1806, everything started talking again about the war against Napoleon with greater heat, lower than the last fate. Appointments buv not only recruited a recruit, but still 9 warriors out of a thousand. Everywhere they cursed Bonaparte with an anathema, and in Moscow there was only a little sense about the upcoming war. For the Rostov family, all the interest is in preparing before the war, only in the fact that Mikolushka is not too good to be left in Moscow and vychikuvav only a tenth of the admission of Denisov in order to go with him at once to the regiment of the saint. The future day not only did not want you to have fun, but still wanted you to the end. Most of the time spent at home, at obidah, evenings and balls.

XI
On the third day of the Feast, Mikola had hurt his home, which for the rest of the hour rarely trotted around with him. This is an official farewell insult, oskolki wines from Denisovym їhav at the regiment after Khreshchennya. Twenty people offended, including Dolokhov and Denisov.
Nikolas in Rostov’s house of love again, the atmosphere of suffocation did not allow themselves to feel such strength, like the day is holy. “Catch the hvilini happily, shake your kohati, kill yourself! Only one and the same help in the world - reshta all the nitty-gritty. I'm just busy here," the atmosphere seemed. Mykola, like and forever, having rolled up two pairs of horses, and then not catching up to visit all the places, de Yomu needed a booty and where Yogo was called, having arrived at home before the insult. As only vin uvishov, vin commemorating and seeing the intensity of the love atmosphere in the booth, and moreover vin commemorating the marvelous zamіshanna that panuє mіzh among the active members of the conspiracy. Particularly blessed were Sonya, Dolokhov, the old countess and the little Natasha. Mykola is sensible, it’s not enough to become obidu between Sonya and Dolokhov, and with the powerful chuynistyu of the heart, it’s even lower and amulets, for an hour of offense, at the occasion of both of them. On the same evening of the third day, holy mav buti one of the quiet balls with Yogel (dance teacher), like wine giving from the saint for all her students, that student.
- Nikolenko, will you go to Yogel? Be kind, go, - Natalya said to you, - asking you especially for wine, and Vasil Dmitritch (tse buv Denisov) is here.
- Wherever I do not go for the order of the ґafіnі!
- When I get it! I celebrated Arkharovim, they have a party, - saying Mykola.
– And you? - I turned back to Dolokhov. I had a good night's sleep, remembering what you didn't need to feed.
- So, maybe ... - Dolokhov coldly and angrily, looking at Sonya and, frowning, looking like that at the club, marveling at P'er, looking again at Mikola.
"It's є," thinking Mikola, and even more affirming himself in his admission to him, that Dolokhov immediately wrote an insult to him. Winning Natalka and asking, what is it?
- And I was kidding you, - said Natasha, sighing up to the new. - I said, you didn’t want to believe everything, - she said triumphantly, - having built up Sonya’s proposition.
Having taken care of Mykola Sonya a lot in the whole hour, even though it didn’t hurt, if you felt guilty. Dolokhov buv is worthy and in deyaky vidnosins a bright party for Sir Sonya. From the look of the old countess, that light could not help you. Before that, I felt like Mikoli, if I felt it, I was embittered against Sonya. Vіn was getting ready to say: “It's miraculous, it made sense, you need to forget the childish obіtsyanki and accept the proposition”; but I didn’t catch the blame more to say.
- You can show yourself! vona led, zowsim led! Natalia spoke up. - Vaughn said, scho to love іnshoy, - she added, troch pomochavsya.
“That way my Sonya couldn’t fix it!” Mykola thought.
- Mom didn’t ask for skilki, she did it, and I know she won’t change what she said ...
- And my mother asked її! - Z dokor said Mikola.
- Yes, - said Natalka. - You know, Nikolenko, don't be angry; but I know that you will not make friends with him. I know, God knows why, I know for sure, you will not make friends.
“Well, you don’t know what,” said Mikola; - Alemeni need to talk to her. What a beauty, Sonya! – adding wine, chuckle.
- Tse such charіvnіst! I will give you її. - I Natalka, having kissed her brother, flowed in.
Through the whiplash, Sonya went away, wicked, that wine was ruined. Mykola pidiishov kissed her hand before her. This was the first time that the stinks on their arrival were talking all the time and about their kohannya.
- Sophie, - having said timidly, and then we will all smile more and more cheerfully, - as if you want to act not only in the bright, in the victorious party; ale vin a beautiful, gentry man ... vin my friend ...
Sonya interrupted yogo.
- I've already taken notice, - she said hastily.
- If you act for me, then I'm afraid that on me ...
Sonya interrupted Yogo again. Vaughn stared at the new one with a blissful, swaying look.
- Nicolas, don't tell me, - said the vagrant.
- No, I'm mushu. Maybe tse suffisance [arrogance] from my side, but to put it more simply. If you follow me, then I am obliged to tell you the whole truth. I love you, I think more...
“I’m old,” said Sonya, having caught herself.
- No, but I’ve been suffocated a thousand times and suffocated, wanting such a sense of friendship, trust, love, I can’t reach anyone, like you. Let's keep me young. Mom doesn't want anything. Well, I just don't blame anything. And I ask you to think about Dolokhov's proposition, - having said wine, forcibly forcing the call of his friend.
- Don't say me. I don't want anything. I love you, like a brother, and forever beloved, and nothing more is required of me.
- Vee angel, I'm not worth you, but I'm less afraid of fooling you. - Mikola once again kissed your hand.

Yogel Mav the most fun Bali in Moscow. This is what the mothers said, marveling at their adolescentes, [girls,] what they say about their own vicheni pas; they spoke and themselves adolescentes and adolescents, [girls and youths,] danced until they dropped; These grown-up girls and young people, like they came to the balls with a thought to go to them and those who know the best fun in them. What fate at these balls became two whores. The two princesses Gorchakov knew the names and went abroad, and they let more people into the glory of the bali. Especially at these balls there were those who didn’t have the master of that gentleman: buv, like fluff flying, following the rules of the art, the good-natured Yogel, who accepted receipts for lessons from all of his guests; it was those who were still waiting for only those who wanted to dance and have fun, as they want 13 and 14-year-old girls, as if they were dressing up old cloth. Moustache, for rare vinnyatki, boules chi were given garnery: so suffocated the stench all laughed and their eyes lit up so much. Some of the best students danced to pas de chale, for the best of Natalka's boulder, as it signified its grace; but at this, the rest of the ball, only ecosesi, anglesi and mazurka danced, which should be fashionable. The hall was taken by Yogel at Bezukhov's booths, and the ball was farther away, as everyone said. There was a lot of beautiful girls, and Rostov pannochki were the best. The stench of offense was especially happy and cheerful. That evening, Sonya, proud of Dolokhov's proposition, of her own explanation of Mikola, was still circling around the house, not letting the maiden smell her scythe, and now she shone brightly with rapturous joy.
Natasha, no less proud of them, that she was in the old cloth, at the right ball, she was more happy. Offenses were in white, muslin cloths with erysipelas stitches.
Natalya became zakokhana from the very same whims, as if she had gone to the ball. Vaughn didn’t succumb to anyone in particular, but she succumbed to everyone. The one at whom the vagrant marveled in that place, as if she marveled, in that vain she was smitten.
- Oh, how good! - All spoke out, podbіgayuchi to Sonia.
Mykola and Denisov walked through the halls, gazing at the dances slyly and patronizingly.
- Like she's sweet, you'll be fine, - Denisov said.
– Who?
- Gafinya Natasha, - Vidpoviv Denisov.
- I’m like a dancer, like a gatz! - Having spoken to the trochs, having said again the wines.
- Who are you talking about?
“About your sister,” Denisov shouted angrily.
Rostov smiled.
- Mon cher comte; Vous etes l'un de mes meilleurs ecoliers, il faut que vous dansiez, said little Yogel, approaching Mikoli. - Voyez combien de jolies demoiselles. good-looking girls!] - In the same way, I turned back to Denisov, also my colossal education.
“Hi, my love, I’m sitting here writing a wall,” Denisov said. - Heba you don't remember how I was rotten with your lessons?
- Oh no! - quickly vtishayuchi yogo, saying Yogel. - For the most part disrespectful, but for the least for the vibes, for so, for the least for the vibes.
They played the mazurka, which was being reintroduced; Mykola did not immediately respond to Yogel and asked for Sonya. Denisov pidsіv to grandma and leaning on the pattern, dulling the beat, schoos merrily reciting and laughing at the old ladies, looking at the young, what they are dancing. Yogel danced with the first couple with Natalka, his pride and the best student. M'yako, gently plucking with his little boots in little boots, Yogel first flew through the hall fearfully, but diligently prepared for Natalka. Denisov did not open his eyes and tapped the beat, with such a look, which clearly showed that he himself did not dance more than what he did not want, but not because he could not. At the middle of the figure, I called Rostov to myself.
- Tsezovsim not those, - having said wine. - Hiba tse Polish mazuka? And a wonderful dance. - Knowing that Denisov and in Poland were famous for his master dance of the Polish mazurka, Mikola went to Natasha:
- Go, choose Denisov. Axis dance! Miracle! - saying wine.
When the blackness of Natasha came again, she stood up and swayed her little bows with her bows, timidly, alone ran through the hall to the kut, where Denisov sat. Vona bachila, so that everyone marvels at her and checks. Mykola Bachiv, that Denisov and Natasha laughed at each other, and that Denisov, laughing radiantly. Vin pidbig.
- Be kind, Vasil Dmitritch, - said Natalka, - let's go, be kind.
- So, scho, vibachte, sir, - saying Denisov.
“Well, wait, Vasya,” said Mikola.
“It’s like they’re wrapping Vaska the cat,” Denisov said in a jarring voice.
- Tsiliy vechir you sleep, - said Natalka.
- Enchantress all z zrob me! - Having said Denisov and squeezing the shablya. Vin viyshov from behind the stiles, mittingly taking his lady by the hand, raising his head and putting his foot forward, checking the beat. Only on horseback and in Mazury you can’t see Denisov’s small stature, and he was such a fine fellow, feeling like wine himself. Seeing the tact, win from the side, it is possible and hot, looking at your lady, uncontrollably tapping with one foot, like a ball, springing up in the air and flying the bridle on the stake, pulling your lady behind you. Vіn did not seem to fly half the hall on one leg, and, it seemed, without looking at the stiles that stood in front of him, and straight rushing at them; ale raptom, clattering his spurs and spreading his legs, zapping on the pedestals, standing like that for a second, with a gurgle of spurs clattering in one place with his feet, spinning swiftly and, with his left foot kicking right, flying again on the stake. Natasha guessed at those who stumbled upon the mother, and she herself did not know how, stalked after him - giving way to you. Now spinning її, then on the right, then on the left hand, then falling on your knees, circling її around yourself, and hopping again and starting forward with such a strimkistyu, nebi vin mav namir, without taking a breath, jump over the mustache of the stone; then raptom again sounded and again robbed new and unsettled knees. If wine, chewing the lady in front of the її mist, clattering with a spur, bowing before her, Natasha did not send him to you. Vaughn stared at the new eye, laughing, not knowing anything. - What is it? Vaughn proclaimed.
Irrespective of those that Yogel did not recognize the sound of the mazurka of the right, everyone was buried by Denisov’s master, they began to vibrate incessantly, and the old ones, smiling, began to talk about Poland and about the old hour. Denisov, reddened in the mazurka and wiped with harshness, went to Natasha and the whole ball did not come out of her.

The display function of the speech change (with a positive substantiation) shows a sprat of primaries. First, for natural values ​​- like a dobutok of equal spivmultipliers. Then the value expands on the number of negative and non-zero values ​​for the rules. In the distance, shotgun displays are seen, for which the values ​​of the display function are dependent on additional roots: . For irrational meanings, the meaning of the assignment is related to the basic concepts of mathematical analysis - to the boundary transition, to the mirroring of the uninterrupted. All the tests are not zastosovnі before trying to expand the display function on the complex value of the indicator, and that, for example, is absolutely unreasonable.

The first step with a complex indicator with a natural basis was introduced by Euler on the basis of an analysis of the low motives of the integral calculation. In some cases, similar lines of algebra, when integrated, give different differences:

At the same time, here another integral formally exits from the first one when replaced by

Zvіdsi can zrobiti vysnovok, scho with a well-defined display function with a complex indicator of turning trigonometric functions of sporidnі with logarithms, and by the same token, the display function is connected with trigonometric ones.

Euler had a lot of humility and fantasy to give a reasonable designation for the show function with the basis.

Because of the appointment, and to that the formula is not brought up, it is possible to just shukat arguments on the cruelty of the intelligence and dozilnosti of such a appointment. Mathematical analysis is already rich in arguments of this kind. We get along with more than one.

Vіdomo, scho s speech mіsce borderline spіvvіdnoshennia: . The right side has a polynomial that has sense i for complex values ​​for . Between sequences of complex numbers are designated as a natural order. The sequence is considered similar, so that the sequence of speech and verbal parts converges and is accepted

We know. For which animal we move to a trigonometric form, moreover, for the argument, it is possible to choose the value of the gap. For such a choice, it is clear that. Dali,

For the boundary transition, it is necessary to change the line in between for and to know the distance between. It is clear that

Father, at the viraz

the speech part is pragne to, it is obvious - to so what

This clumsy mirkuvannya gives one of the arguments on the mediocrity of Euler's designation of the show function.

Let us now establish that the multiplied values ​​of the display function are added together. Diyno:

2. Euler formulas.

Let's do it with the assigned display function. We take:

Replacing b with -b is taken away

Adding and seeing term by term qi equal, we know the formulas

called the Euler formulas. The stench establishes a link between trigonometric functions and show and show signs.

3. Natural logarithm of a complex number.

A complex number given in trigonometric form can be written in the form of a complex number. She takes all the good power of the trigonometric form, and is still short. Further, it is natural to consider that the logarithm of the th modulus is shown by the verbal partial logarithm of a complex number, and the th argument is manifested by the verbal partial logarithm. With the whole world I explain the “logarithmic” power of the argument - the argument to the creation of a rich sum of arguments in spivmulniki.

Plan:

Entry

• 1 speech logarithm
  • 1.1 Power
  • 1.2 logarithmic function
  • 1.3 Natural logarithms
  • 1.4 Decimal logarithms
• 2 Complex logarithm
  • 2.1 Appointment of that power
  • 2.2 Apply
  • 2.3 Analytical follow-up
  • 2.4 Riemann surface
• 3 historical drawing
  • 3.1 speech logarithm
  • 3.2 Complex logarithm
• 4 Logarithmic tables
• 5 programs
Literature
Notes

Entry

Rice. 1. Graphs of logarithmic functions

Logarithm of a number b on a stand a (Type Greek. λόγος - "word", "setting" that ἀριθμός - “number”) is designated as an indicator of a step, in which it is necessary to call a base a, to take the number b. Designation: . It follows that the records are equally strong.

For example, to that.

1. Speech logarithm

Logarithm of speech number log a b max sens at . Apparently, the display function y = a x monotonous and dermal meaning nabuy only once, moreover, the range of її znachen to avenge all positive speech numbers. It is clear that the meaning of the verbal logarithm of a positive number is always clear and unambiguous.

The widest zastosuvannya knew this kind of logarithms.

1.1. power

proof

Let's say what.

(scratches for the mind bc > 0). ■

proof

Let us know that

(shards for the mind ■

proof

Vikoristovuєmo to prove the identity. The logarithm of the insult to the part of the sameness on the substituent with. We take:

proof

Let's say what.

(so like b p> 0 for the mind). ■

proof

Let us know that

proof

logarithm to left and right of the base c :

Left part: Rights of part:

The jealousy of viraz is obvious. logarithms are equal, then, due to the monotonicity of the logarithmic function, equal and themselves are different. ■

1.2. logarithmic function

How to look at a number that is logarithmic as a change, we take it logarithmic function y= log a x (Div. Fig. 1). Vaughn is credited for . Value area: .

Function є strictly growing at a> 1 i strictly decreasing at 0< a < 1 . График любой логарифмической функции проходит через точку (1;0) . Функция непрерывна и неограниченно дифференцируема всюду в своей области определения.

Straight x= 0 є left vertical asymptote, scaling at a> 1 ta at 0< a < 1 .

Pokhіdna logarithmіchnіchnіchnіchnіchnіchnіchn_chn_chn_chn_chna_dvor_vnyuє:

proof

I. We know that

Let's write down the identity e ln x = x i prodifferentiyuemo yoga lion and right part

We take it, sho, the stars are screaming, sho

II. Let us know that

The logarithmic function creates isomorphism between the multiplicative group of positive real numbers and the additive group of all speech numbers.

1.3. Natural logarithms

Zvyazok іz tenth logarithm: .

As indicated above, for a similar natural logarithm, a simple formula is valid:

Therefore, in mathematical studies, it is more important to win the natural logarithm itself. The stench often blames the hour of the difference in differential equivalences, according to statistical fallows (for example, the division of prime numbers) too thinly.

It is easy to know the inconsistencies of the integral in the natural logarithm by integrating the parts:

The layout in the Taylor series can be represented by the following order:
with fair equality

(1)

Zokrema,

This series converges more, moreover, the left part of the formula can now be used to form the logarithm of any positive number.

1.4. Decimal logarithms

Rice. 2a. logarithmic scale

Rice. 2b. Logarithmic scale with values

Logarithms on base 10 (sign: lg a) before the wine run, calculators were widely stoked for calculation. Non-uniform scale of tens logarithms should be applied to logarithmic lines. A similar scale is victorious in the rich circles of science, for example:

• Physics - sound intensity (decibels).
• Astronomy is a scale of radiance of stars.
• Chemistry - activity of water ions (pH).
• Seismology - Richter scale.
• The theory of music is a musical scale, a hundred frequencies of musical sounds.
• History is a logarithmic time scale.

The logarithmic scale is also widely used for showing the indicator of the step in static fallows and the coefficient for the indicator of the exponent. With this graph, prompting on a logarithmic scale along one or two axes, looks straight, more simple to follow.

2. Complex logarithm

2.1. Appointment of that power

For complex numbers, the logarithm is used as i, as speech. In practice, it is possible to use the natural complex logarithm, which is significant and significant as the impersonal of all complex numbers z such that e z = w . The complex logarithm is used for whatever, and the first part of speech is unambiguous, in that hour it may have an impersonal meaning. This is why yoga is called a rich-valued function. How to reveal w in the display form:

,

then the logarithm is behind the formula:

Here is the speech logarithm, r = | w | , k- quite a number. The value that is taken at k= 0 is called leading values complex natural logarithm; it is customary to take a new value of the argument in the interval (− π,π] . head gulka logarithm i is assigned. Sometimes they also indicate the value of the logarithm, which does not lie on the head of the head.

Three formulas are obvious:

• The speech range of the logarithm is assigned to the formula:

• The logarithm of a large number is behind the formula:

Oskilki complex trigonometric functions related to the exponential (Euler's formula), the complex logarithm as a reversal to the exponential function of reciprocals to the reversal trigonometric functions. An example of such a call:

2.2. Apply

Let's bring the logarithm for the last arguments:

Let's keep it safe when converting complex logarithms, vrakhovuyuchi, because the smells are rich, and therefore, the equality of the logarithms of any virus does not show the equality of these viruses. An example of a pardon mirkuvannya:

iπ = log(−1) = log((− i) 2) = 2ln(− i) = 2(− iπ / 2) = − iπ - obvious stupidity.

It is significant that the left-hander is worth the head value of the logarithm, and the right-hander is the value of the lower head ( k= − 1). The reason for the pardon is the carelessness of victoriousness of power, like, in a flash of seeming, may be on the verge of a complex change, the entire set of values ​​of the logarithm, and not just a smut value.

2.3. Analytical follow-up

Rice. 3. Complex logarithm (visible part)

The logarithm of a complex number can also be used as an analytical continuation of the speech logarithm over the entire complex plane. Let the curve Γ start in one, do not pass through zero and do not change the negative part of the speech axis. Same smut value of the logarithm in final points w the curve Γ can be taken from the formula:

If Γ is a simple curve (without self-permutation), then for the numbers that lie on it, the logarithmic identity can be fixed without a fight, for example

If you allow the curve Γ to transpose the negative part of the speech axis, then the first such transition is to transfer the result from the neck of the head value to the side of the tongue, and the skin advances the transition to the left side of the tongue, similarly to the shift along the tongues of the logarithmic function (div. babies).

3 formulas for analytical follow-up, what should be the logarithm

For whatever stake S, which wraps point 0:

The integral is taken from the positive direction (anti-Year arrow). The value lies at the basis of the theory of recovery.

It is also possible to assign an analytic continuation of the complex logarithm after the addition of the series (1) induced above, which is further developed by the complex argument. Prote z view of the layout of the following, scho for unity of wines is equal to zero, so that the series lie more than the head of the rich-valued function of the complex logarithm.

2.4. Riemann surface

Complex logarithmic function - an example of a Riemann surface; The visible part (Fig. 3) is made up of an infinite number of needles twisted like a spiral. Tsya surface is single-linked; єє single zero (of the first order) go beyond z= 1 , special points: z= 0 (points of uncompromising order).

The Riemann surface of the logarithm is the universal curve for the complex plane without the point 0 .

3. Historical drawing

3.1. speech logarithm

The demand for folding roses in the 16th century was growing rapidly, and significant parts of the difficulty in arranging for multiplications and rozpodil of rich-valued numbers, and to bring to the roots. For example, a hundred mathematicians, maybe at once, fell asleep on the idea of ​​​​an idea: replace the labor worker with a simple addition, setting up a special table of geometrical and arithmetic progressions for help, with which it will be geometrical. Todi and rozpodil are automatically replaced by an immeasurably simpler and more obvious, and the root of the step n be reduced to the sub-logarithm of the sub-root virase on n. First I published the idea in my book “ Arithmetica integra Michael Shtifel, who, vtim, did not report serious efforts for the implementation of his ideas.

In 1614, the Scottish mathematician-amateur John Napier published his Latin twirl under the title " Description of the amazing table of logarithms"(Lat. Mirifici Logarithmorum Canonis Descriptio ). A new word has a short description of the logarithms of those powers, as well as 8-digit tables of the logarithms of sines, cosines and tangents, with the term 1. logarithm, Proponation by Napier, having established himself in science. Theory of logarithms Neper viklav in the next book " Pobudov amazing tables of logarithms"(Lat. Mirifici Logarithmorum Canonis Constructio ), seen posthumously in 1619 roci yogo synom.

The understanding of the function was not the same, and Neper calculated the logarithm kinematically, setting equal and logarithmic improvements; for example, the logarithm of the sine of vin was designated as follows:

The logarithm of this sine is a number, as if arithmetically it grew up with the same swidkistyu, with whichever sine began to change geometrically.

For current values, the kinematic Napier model can be represented by differential equations: dx/x=-dy/M de M - scaling multiplier, introductions for the fact that the value was a whole number with the necessary number of signs (decimal fractions also did not know a wide range). Napier taking M=10000000.

Strictly kazhuchi, Neper tabulating the wrong function, as if at once called the logarithm. In order to define the LogNap(x) function, it is related to the natural logarithm as follows:

Obviously, LogNap(M) = 0, so the logarithm of the "full sine" є zero - tsgogo i Domagavsya Neper to its destination. .

The main power of Napier's logarithm: just as magnitudes make a geometric progression, their logarithms make an arithmetic progression. Prote the rules of the logarithm for the non-Peer function were revised in the same way as the rules of the daily logarithm.

For example, LogNap(ab) = LogNap(a) + LogNap(b) - LogNap(1).

It's a pity that the values ​​of Neper's tables missed the calculation of the pardon after the sixth sign. However, the new methodology for calculating the greatest popularity did not work, and a lot of European mathematicians, including Kepler, took up the folding of logarithmic tables. Already after 5 years, 1619, London teacher of mathematics John Spydell ( John Spidell) having seen Napier's tables, the reworked ones actually became tables of natural logarithms (when scaling to whole numbers Spydell took). The term "natural logarithm" was coined by the Italian mathematician P'etro Mengoli. Pietro Mengoli)) in the middle of the XVI century.

In the 1620s, Edmund Wingate and William Wydred blamed the first logarithmic line, before the advent of intestinal calculators - an indispensable tool for an engineer.

Close to the daily understanding of the logarithm - like an operation, a turning point in the foot - first appeared in Wallis and Johann Bernoulli, and the rest was legitimized by Euler in the XVIII century. In the book “Introduction to the analysis of inconsistencies” (1748), Euler gave a modern designation of both display and logarithmic functions, navіv razladannja їх in static lavi, especially calling on the role of the natural logarithm.

Euler should be able to extend the logarithmic function to the complex domain.

3.2. Complex logarithm

The first to try to solve logarithms for complex numbers at the turn of the XVII-XVIII century, Leibnitz and Johann Bernoulli, however, did not succeed in creating a coherent theory - we had to face these reasons, that even the most understandable logarithm was not yet clearly assigned. The discussion on this subject was carried on between Leibniz and Bernoulli, and in the middle of the 18th century between d'Alembert and Euler. Bernoulli and d'Alembert respected what should be log(-x) = log(x). The new theory of the logarithms of negative and complex numbers was published by Euler in 1747-1751, but in fact it does not matter at all in modern times.

Wanting a super girl trival (d'Alembert, having defended his own thought and reportedly arguing in the articles of his Encyclopedia and other practical work), Euler's thought quickly took away the blatant exclamation.

4. Logarithmic tables

Logarithmic tables

From the powers of the logarithm, we see that instead of the laborious multiplicity of richly significant numbers, we need to know (behind the tables) and add up their logarithms, and then we will use the tables themselves to viconate the potency, to know the value of the result from the th logarithm. Vikonanny rozpodіlu vіdrіznyaetsya less than tim, scho logarithms vіdnіmayutsya. Laplace said that the wines of logarithms "continued the life of astronomers", speeding up the calculation process in a big way.

When transferring ten Komi in number to n the value of the tenth logarithm of the number is changed to n. For example, lg8314.63 = lg8.31463+3. Sounds like it's enough to put together a table of tenth logarithms for numbers in the range from 1 to 10.

The first tables of logarithms were published by John Napier (1614), and the stench avenged only the logarithms of trigonometric functions, moreover, with pardons. Independently of his own tables, he published Іost Bürgі, a friend of Kepler (1620). In 1617, the Oxford professor of mathematics Henry Briggs published tables, which already included dozens of logarithms of the numbers themselves, from 1 to 1000, with 8 (last - from 14) signs. But pardons appeared in the tables of Briggs. The first bezpomilkovo vidannya on the basis of the table Vega (1783) appeared only in 1857 in Berlin (table Bremiver).

In Russia, the first tables of logarithms were seen in 1703 with the participation of L. F. Magnitsky. The SRSR issued a collection of logarithm tables.

• Bradis V. M. Chotyric mathematical tables. 44th edition, M., 1973.

The tables of Bradys (1921) were victorious at the headings and in the engineering departments, which did not require great accuracy. The stench avenged the mantis of tenth logarithms of numbers and trigonometric functions, natural logarithms and decimals of other coordinated rozrahunkov's tools.

• Vega G. Tables of seven-digit logarithms, 4 editions, M., 1971.

Professional selection for accurate calculations.

• Five-digit tables of natural values ​​of trigonometric quantities, їх logarithms and logarithms of numbers, 6th edition, M.: Nauka, 1972.
• Tables of natural logarithms, 2nd edition, in 2 volumes, M: Nauka, 1971.

In this hour, with the expansion of calculators, it is necessary to use a table of logarithms.

M, Specificity (complex analysis).

The main power of the logarithm, the graph of the logarithm, the range of designation, the anonymous value, the main formulas, and the growth rate of the fall are given. The meaning of the similar logarithm is examined. And so is the integral, arranging in a series of states, that manifestation behind the help of complex numbers.

Zmist

Significance area, impersonal meaning, growth, decline

The logarithm is a monotonic function, so there are no extremums. The main power of the logarithm is presented in the tables.

Appointment area	0 < x < + ∞	0 < x < + ∞
area value	- ∞ < y < + ∞	- ∞ < y < + ∞
monotony	monotonously growing	change monotonously
Zero, y = 0	x= 1	x= 1
Knots peretina z vіssyu ordinates, x = 0	no	no
	+ ∞	- ∞
	- ∞	+ ∞

Private values

The logarithm on the base 10 is called tenth logarithm i is denoted like this:

logarithm on base e called natural logarithm:

Basic formulas for logarithms

The power of the logarithm, which is exhaled from the designation of the pivotal function:

The main power of logarithms and yogo traces

Base Change Formula

Logarithm is a mathematical operation of taking a logarithm. When taking a logarithm, the creation of multipliers is transformed into a sum of terms.
Potentialization is a mathematical operation, reversible logarithm. With potentiation, the foundation is set to be built at the feet of the viraz, over which the potentiation is victorious. At the same time, the sum of the members are transformed into the creation of partners.

Proof of the basic formulas of logarithms

Formulas associated with logarithms are derived from the formulas for display functions and the designation of the reversal function.

Let's look at the power of the display function
.
Todi
.
It is necessary to determine the quality of the display function
:
.

We bring the formula for replacing the base.
;
.
However, c = b may be:

Return function

The return function for the logarithm on the basis of a is the display function with the indicator of the step a.

Yakscho something

Yakscho something

Pokhіdna logarithm

Pokhіdna logarithm vіd modulo x:
.
Pokhіdna n-th order:
.
Visnovok formulas > > >

To understand the similar logarithm of yoga, it is necessary to bring it to the foundation e.
;
.

Integral

The integral in the form of a logarithm is calculated by integrating parts: .
Otzhe,

Virazi through complex numbers

Let's look at the function of a complex number z:
.
Virazimo complex number z via module r that argument φ :
.
Todi, vikoristovuyuchi power logarithm, maybe:
.
Abo

Prote, argument φ not clearly defined. Put Yakshcho
de n - qile,
then it will be one and the same number for different n.

Therefore, the logarithm, as a function of a complex zminny, is not a single-valued function.

Arrangement in stacked row

At the time of laying out:

Wikoristan literature:
I.M. Bronstein, K.A. Semendyaev, Mathematics guide for engineers and university students, Lan, 2009.

Div. also: