Definition and properties

Complex zero does not have a logarithm because the complex exponent does not take the value zero. Non-zero texvc can be represented in demonstrative form:

Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): z=r \cdot e^(i (\varphi + 2 \pi k))\;\;, Where Unable to parse expression (Executable file texvc not found; See math/README for setup help.): k- arbitrary integer

Then Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \mathrm(Ln)\,z is found by the formula:

Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): \mathrm(Ln)\,z = \ln r + i \left(\varphi + 2 \pi k \right)

Here Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \ln\,r= \ln\,|z|- real logarithm. It follows from this:

It is clear from the formula that one and only one of the values ​​has an imaginary part in the interval Unable to parse expression (Executable file texvc . This value is called main importance complex natural logarithm. The corresponding (already unambiguous) function is called main branch logarithm and is denoted Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \ln\,z. Sometimes through Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \ln\, z also denote the value of the logarithm that does not lie on the main branch. If Unable to parse expression (Executable file texvc not found; See math/README for setup help.): z is a real number, then the principal value of its logarithm coincides with the ordinary real logarithm.

From the above formula it also follows that the real part of the logarithm is determined as follows through the components of the argument:

Unable to parse expression (Executable file texvc not found; See math/README for help with setup.): \operatorname(Re)(\ln(x+iy)) = \frac(1)(2) \ln(x^2+y^2)

The figure shows that the real part as a function of the components is centrally symmetric and depends only on the distance to the origin. It is obtained by rotating the graph of the real logarithm around the vertical axis. As it approaches zero, the function tends to Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): -\infty.

The logarithm of a negative number is found by the formula:

Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \mathrm(Ln) (-x) = \ln x + i \pi (2 k + 1) \qquad (x>0,\ k = 0, \pm 1 ,\pm 2\dots)

Examples of complex logarithm values

Let us present the main value of the logarithm ( Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \ln) and its general expression ( Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \mathrm(Ln)) for some arguments:

Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \ln (1) = 0;\; \mathrm(Ln) (1) = 2k\pi i Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \ln (-1) = i \pi;\; \mathrm(Ln) (-1) = (2k+1)i \pi Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): \ln (i) = i \frac(\pi) (2);\; \mathrm(Ln) (i) = i \frac(4k+1)(2) \pi

You should be careful when converting complex logarithms, taking into account that they are multi-valued, and therefore the equality of the logarithms of any expressions does not imply the equality of these expressions. Example erroneous reasoning:

Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): i\pi = \ln(-1) = \ln((-i)^2) = 2\ln(-i) = 2(-i\pi/2 ) = -i\pi- an obvious mistake.

Note that on the left is the main value of the logarithm, and on the right is the value from the underlying branch ( Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): k=-1). The cause of the error is careless use of the property Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \log_a((b^p)) = p~\log_a b, which, generally speaking, implies in the complex case the entire infinite set of values ​​of the logarithm, and not just the main value.

Complex logarithmic function and Riemann surface

Due to its simply connectedness, the Riemann surface of the logarithm is a universal covering for the complex plane without a point Unable to parse expression (Executable file texvc .

Analytical continuation

The logarithm of a complex number can also be defined as the analytic continuation of the real logarithm to the entire complex plane. Let the curve Unable to parse expression (Executable file texvc starts at one, does not go through zero and does not cross the negative part of the real axis. Then the principal value of the logarithm at the end point Unable to parse expression (Executable file texvc not found; See math/README for setup help.): w crooked Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \Gamma can be determined by the formula:

Unable to parse expression (Executable file texvc not found; See math/README for help with setting up.): \ln z = \int\limits_\Gamma (du \over u)

If Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \Gamma- a simple curve (without self-intersections), then for numbers lying on it, logarithmic identities can be used without fear, for example:

Unable to parse expression (Executable file texvc not found; See math/README for help with setup.): \ln (wz) = \ln w + \ln z, ~\forall z,w\in\Gamma\colon zw\in \Gamma

The main branch of the logarithmic function is continuous and differentiable on the entire complex plane, except for the negative part of the real axis, on which the imaginary part changes abruptly to Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): 2\pi. But this fact is a consequence of the artificial limitation of the imaginary part of the main value by the interval Unable to parse expression (Executable file texvc not found; See math/README for setup help.): (-\pi, \pi]. If we consider all branches of the function, then continuity occurs at all points except zero, where the function is not defined. If you resolve the curve Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \Gamma cross the negative part of the real axis, then the first such intersection transfers the result from the main value branch to the adjacent branch, and each subsequent intersection causes a similar shift along the branches of the logarithmic function (see figure).

From the analytic continuation formula it follows that on any branch of the logarithm:

Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \frac(d)(dz) \ln z = (1\over z)

For any circle Unable to parse expression (Executable file texvc not found; See math/README for setup help.): S, covering the point Unable to parse expression (Executable file texvc not found; See math/README for setup help.): 0 :

Unable to parse expression (Executable file texvc not found; See math/README for help with setting up.): \oint\limits_S (dz \over z) = 2\pi i

The integral is taken in the positive direction (counterclockwise). This identity underlies the theory of residues.

One can also define the analytic continuation of the complex logarithm using series known for the real case:

However, from the form of these series it follows that at one the sum of the series is equal to zero, that is, the series relates only to the main branch of the multivalued function of the complex logarithm. The radius of convergence of both series is 1.

Connection with inverse trigonometric and hyperbolic functions

Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \operatorname(Arcsin) z = -i \operatorname(Ln) (i z + \sqrt(1-z^2)) Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \operatorname(Arccos) z = -i \operatorname(Ln) (z + i\sqrt(1-z^2)) Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): \operatorname(Arctg) z = -\frac(i)(2) \ln \frac(1+z i)(1-z i) + k \pi \; (z \ne \pm i) Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): \operatorname(Arcctg) z = -\frac(i)(2) \ln \frac(z i-1)(z i+1) + k \pi \; (z \ne \pm i) Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \operatorname(Arsh)z = \operatorname(Ln)(z+\sqrt(z^2+1))- inverse hyperbolic sine Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): \operatorname(Arch)z=\operatorname(Ln) \left(z+\sqrt(z^(2)-1) \right)- inverse hyperbolic cosine Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \operatorname(Arth)z=\frac(1)(2)\operatorname(Ln)\left(\frac(1+z)(1-z)\right)- inverse hyperbolic tangent Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \operatorname(Arcth)z=\frac(1)(2)\operatorname(Ln)\left(\frac(z+1)(z-1)\right)- inverse hyperbolic cotangent

Historical sketch

The first attempts to extend logarithms to complex numbers were made at the turn of the 17th-18th centuries by Leibniz and Johann Bernoulli, but they failed to create a holistic theory, primarily because the very concept of logarithm was not yet clearly defined. The discussion on this issue took place first between Leibniz and Bernoulli, and in the middle of the 18th century between D’Alembert and Euler. Bernoulli and D'Alembert believed that it should be determined Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \log(-x) = \log(x), while Leibniz proved that the logarithm of a negative number is an imaginary number. The complete theory of logarithms of negative and complex numbers was published by Euler in 1747-1751 and is essentially no different from the modern one. Although the debate continued (D'Alembert defended his point of view and argued it in detail in an article in his Encyclopedia and in other works), Euler's approach received universal recognition by the end of the 18th century.

Write a review about the article "Complex logarithm"

Literature

Theory of logarithms

• Korn G., Korn T.. - M.: Nauka, 1973. - 720 p.
• Sveshnikov A. G., Tikhonov A. N. Theory of functions of a complex variable. - M.: Nauka, 1967. - 304 p.
• Fikhtengolts G. M. Course of differential and integral calculus. - ed. 6th. - M.: Nauka, 1966. - 680 p.

History of logarithms

• Mathematics XVIII century// / Edited by A. P. Yushkevich, in three volumes. - M.: Science, 1972. - T. III.
• Kolmogorov A. N., Yushkevich A. P. (eds.). Mathematics of the 19th century. Geometry. Theory of analytic functions. - M.: Science, 1981. - T. II.

Notes

1. Logarithmic function. // . - M.: Soviet Encyclopedia, 1982. - T. 3.
2. , Volume II, pp. 520-522..
3. , With. 623..
4. , With. 92-94..
5. , With. 45-46, 99-100..
6. Boltyansky V. G., Efremovich V. A.. - M.: Nauka, 1982. - P. 112. - (Kvant Library, issue 21).
7. , Volume II, pp. 522-526..
8. , With. 624..
9. , With. 325-328..
10. Rybnikov K. A. History of mathematics. In two volumes. - M.: Publishing house. Moscow State University, 1963. - T. II. - P. 27, 230-231..
11. , With. 122-123..
12. Klein F.. - M.: Science, 1987. - T. II. Geometry. - pp. 159-161. - 416 s.

An excerpt characterizing the Complex logarithm

From the wild horror that gripped us, we rushed like bullets across a wide valley, not even thinking that we could quickly go to another “floor”... We simply did not have time to think about it - we were too scared.
The creature flew right above us, loudly clicking its gaping toothy beak, and we rushed as fast as we could, splashing vile slimy splashes to the sides, and mentally praying that something else would suddenly interest this creepy “miracle bird”... It was felt. that she was much faster and we simply had no chance to break away from her. As luck would have it, not a single tree grew nearby, there were no bushes, or even stones behind which one could hide, only an ominous black rock could be seen in the distance.
- There! – Stella shouted, pointing her finger at the same rock.
But suddenly, unexpectedly, right in front of us, a creature appeared from somewhere, the sight of which literally froze our blood in our veins... It appeared as if “straight out of thin air” and was truly terrifying... The huge black carcass was completely covered long, coarse hair, making him look like a pot-bellied bear, only this “bear” was as tall as a three-story house... The monster’s lumpy head was “crowned” with two huge curved horns, and the eerie mouth was decorated with a pair of incredibly long fangs, sharp as knives, just by looking to which, with fright, our legs gave way... And then, incredibly surprising us, the monster easily jumped up and... picked up the flying “muck” on one of its huge fangs... We froze in shock.
- Let's run!!! – Stella squealed. – Let’s run while he’s “busy”!..
And we were ready to rush again without looking back, when suddenly a thin voice sounded behind our backs:
- Girls, wait!!! No need to run away!.. Dean saved you, he is not an enemy!
We turned around sharply - a tiny, very beautiful black-eyed girl was standing behind us... and was calmly stroking the monster that had approached her!.. Our eyes widened in surprise... It was incredible! Certainly - it was a day of surprises!.. The girl, looking at us, smiled welcomingly, not at all afraid of the furry monster standing next to us.
- Please don't be afraid of him. He is very kind. We saw that Ovara was chasing you and decided to help. Dean was great, he made it on time. Really, my dear?
“Good” purred, which sounded like a slight earthquake, and, bending his head, licked the girl’s face.
– Who is Owara, and why did she attack us? – I asked.
“She attacks everyone, she’s a predator.” And very dangerous,” the girl answered calmly. – May I ask what you are doing here? You're not from here, girls?
- No, not from here. We were just walking. But the same question for you - what are you doing here?
“I’m going to see my mother...” the little girl became sad. “We died together, but for some reason she ended up here.” And now I live here, but I don’t tell her this, because she will never agree with it. She thinks I'm just coming...
– Isn’t it better to just come? It’s so terrible here!.. – Stella shrugged her shoulders.
“I can’t leave her here alone, I’m watching her so that nothing happens to her.” And here Dean is with me... He helps me.
I just couldn’t believe it... This little brave girl voluntarily left her beautiful and kind “floor” to live in this cold, terrible and alien world, protecting her mother, who was very “guilty” in some way! I don’t think there would be many people so brave and selfless (even adults!) who would dare to undertake such a feat... And I immediately thought - maybe she just didn’t understand what she was going to doom herself to?!
– How long have you been here, girl, if it’s not a secret?
“Recently...” the black-eyed baby answered sadly, tugging at a black lock of her curly hair with her fingers. - I got into this beautiful world when she died!.. He was so kind and bright!.. And then I saw that my mother was not with me and rushed to look for her. It was so scary at first! For some reason she was nowhere to be found... And then I fell into this terrible world... And then I found her. I was so scared here... So lonely... Mom told me to leave, she even scolded me. But I can’t leave her... Now I have a friend, my good Dean, and I can already somehow exist here.
Her “good friend” growled again, which gave Stella and me huge “lower astral” goosebumps... Having collected myself, I tried to calm down a little and began to take a closer look at this furry miracle... And he, immediately feeling that he was noticed, he terribly bared his fanged mouth... I jumped back.
- Oh, don't be afraid, please! “He’s smiling at you,” the girl “reassured.”
Yeah... You'll learn to run fast from such a smile... - I thought to myself.
- How did it happen that you became friends with him? – Stella asked.
– When I first came here, I was very scared, especially when such monsters as you were attacking today. And then one day, when I almost died, Dean saved me from a whole bunch of creepy flying “birds”. I was also scared of him at first, but then I realized what a heart of gold he has... He is the most best friend! I never had anything like this, even when I lived on Earth.
- How did you get used to it so quickly? His appearance is not quite, let’s say, familiar...
– And here I understood one very simple truth, which for some reason I did not notice on Earth - appearance does not matter if a person or a creature has kind heart... My mother was very beautiful, but at times she was very angry too. And then all her beauty disappeared somewhere... And Dean, although scary, is always very kind, and always protects me, I feel his kindness and am not afraid of anything. But you can get used to the appearance...
– Do you know that you will be here for a very long time, much longer than people live on Earth? Do you really want to stay here?..
“My mother is here, so I have to help her.” And when she “leaves” to live on Earth again, I will also leave... To where there is more goodness. In that scary world and people are very strange - as if they don’t live at all. Why is that? Do you know anything about this?
– Who told you that your mother would leave to live again? – Stella became interested.
- Dean, of course. He knows a lot, he’s lived here for a very long time. He also said that when we (my mother and I) live again, our families will be different. And then I won’t have this mother anymore... That’s why I want to be with her now.
- How do you talk to him, your Dean? – Stella asked. – And why don’t you want to tell us your name?
But it’s true – we still didn’t know her name! And they didn’t know where she came from either...
– My name was Maria... But does that really matter here?
- Surely! – Stella laughed. - How can I communicate with you? When you leave, they will give you a new name, but while you are here, you will have to live with the old one. Did you talk to anyone else here, girl Maria? – Stella asked, jumping from topic to topic out of habit.
“Yes, I talked...” the little girl said hesitantly. “But they are so strange here.” And so unhappy... Why are they so unhappy?
– Is what you see here conducive to happiness? – I was surprised by her question. – Even the local “reality” itself kills any hopes in advance!.. How can you be happy here?
- Don't know. When I’m with my mother, it seems to me that I could be happy here too... True, it’s very scary here, and she really doesn’t like it here... When I said that I agreed to stay with her, she yelled at me and said that I’m her “brainless misfortune”... But I’m not offended... I know that she’s just scared. Just like me...
– Perhaps she just wanted to protect you from your “extreme” decision, and only wanted you to go back to your “floor”? – Stella asked carefully, so as not to offend.
– No, of course... But thank you for the good words. Mom often called me something else good names, even on Earth... But I know that this is not out of anger. She was simply unhappy that I was born, and often told me that I ruined her life. But it wasn't my fault, was it? I always tried to make her happy, but for some reason I wasn’t very successful... And I never had a dad. – Maria was very sad, and her voice was trembling, as if she was about to cry.
Stella and I looked at each other, and I was almost sure that similar thoughts visited her... I already really didn’t like this spoiled, selfish “mother”, who, instead of worrying about her child herself, did not care about his heroic sacrifice at all I understood and, in addition, I also hurt her painfully.
“But Dean says that I’m good, and that I make him very happy!” – the little girl babbled more cheerfully. “And he wants to be friends with me.” And others I've met here are very cold and indifferent, and sometimes even evil... Especially those who have monsters attached...
“Monsters—what?..” we didn’t understand.
- Well, they have terrible monsters sitting on their backs and telling them what they must do. And if they don’t listen, the monsters mock them terribly... I tried to talk to them, but these monsters won’t allow me.
We understood absolutely nothing from this “explanation,” but the very fact that some astral beings were torturing people could not remain “explored” by us, so we immediately asked her how we could see this amazing phenomenon.
- Oh, yes everywhere! Especially at the “black mountain”. There he is, behind the trees. Do you want us to go with you too?
- Of course, we will be only too happy! – the delighted Stella immediately answered.
To be honest, I also didn’t really smile at the prospect of dating someone else, “creepy and incomprehensible,” especially alone. But interest overcame fear, and we, of course, would have gone, despite the fact that we were a little afraid... But when such a defender as Dean walked with us, it immediately became more fun...
And then, after a short moment, real Hell unfolded before our eyes, wide open with amazement... The vision was reminiscent of the paintings of Bosch (or Bosc, depending on what language you translate it into), a “crazy” artist who once shocked the whole world with his art world... He, of course, was not crazy, but was simply a seer who for some reason could only see the lower Astral. But we must give him his due - he portrayed him superbly... I saw his paintings in a book that was in my dad’s library, and I still remembered the eerie feeling that most of his paintings carried...
“What a horror!..” whispered the shocked Stella.
One could probably say that we have already seen a lot here, on the “floors”... But even we were not able to imagine this in our most terrible nightmare!.. Behind the “black rock” something completely opened up unthinkable... It looked like a huge, flat “cauldron” carved into the rock, at the bottom of which crimson “lava” was bubbling... The hot air “burst” everywhere with strange flashing reddish bubbles, from which scalding steam burst out and fell in large drops to the ground, or to the people who fell under it at that moment... Heartbreaking screams were heard, but immediately fell silent, as the most disgusting creatures sat on the backs of the same people, who with a contented look “controlled” their victims, not paying the slightest attention to their suffering... Under the naked feet of people, hot stones turned red, the crimson earth, bursting with heat, bubbled and “melted”... Splashes of hot steam burst through huge cracks and, burning the feet of human beings sobbing in pain, were carried into the heights, evaporating with a light smoke ... And in the very middle of the “pit” flowed a bright red, wide fiery river, into which, from time to time, the same disgusting monsters unexpectedly threw one or another tormented entity, which, falling, caused only a short splash of orange sparks, and then but, turning for a moment into a fluffy white cloud, it disappeared... forever... It was real Hell, and Stella and I wanted to “disappear” from there as soon as possible...
“What are we going to do?” Stella whispered in quiet horror. - Do you want to go down there? Is there anything we can do to help them? Look how many there are!..
We stood on a black-brown, heat-dried cliff, observing the “mash” of pain, hopelessness, and violence that stretched below, filled with horror, and felt so childishly powerless that even my militant Stella this time categorically folded her ruffled “wings.” “and was ready at the first call to rush off to her own, so dear and reliable, upper “floor”...

Material from Wikipedia - the free encyclopedia

Definition and properties

Complex zero does not have a logarithm because the complex exponent does not take the value zero. Non-zero $z$ can be represented in demonstrative form:

$z=r\; \backslash cdot\; e^(i\; (\backslash varphi\; +\; 2\; \backslash pi\; k))\backslash ;\backslash ;,$ Where $k$- arbitrary integer

Then $\backslash mathrm(Ln)\backslash ,z$ is found by the formula:

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

Here $\backslash ln\backslash ,r=\; \backslash ln\backslash ,|z|$- real logarithm. It follows from this:

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

Examples of complex logarithm values

Let us present the main value of the logarithm ( $\backslash ln$) and its general expression ( $\backslash mathrm(Ln)$) for some arguments:

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

You should be careful when converting complex logarithms, taking into account that they are multi-valued, and therefore the equality of the logarithms of any expressions does not imply the equality of these expressions. Example erroneous reasoning:

$i\backslash pi\; =\; \backslash ln(-1)\; =\; \backslash ln((-i)^2)\; =\; 2\backslash ln(-i)\; =\; 2(-i\backslash pi/2)\; =\; -i\backslash pi$- an obvious mistake.

Note that on the left is the main value of the logarithm, and on the right is the value from the underlying branch ( $k=-1$). The cause of the error is careless use of the property $\backslash log\_a((b^p))\; =\; p~\backslash log\_a\; b$, which, generally speaking, implies in the complex case the entire infinite set of values ​​of the logarithm, and not just the main value.

Complex logarithmic function and Riemann surface

Due to its simply connectedness, the Riemann surface of the logarithm is a universal covering for the complex plane without a point $0$.

Analytical continuation

The logarithm of a complex number can also be defined as the analytic continuation of the real logarithm to the entire complex plane. Let the curve $\backslash Gamma$ starts at one, does not go through zero and does not cross the negative part of the real axis. Then the principal value of the logarithm at the end point $w$ crooked $\backslash Gamma$ can be determined by the formula:

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

If $\backslash Gamma$- a simple curve (without self-intersections), then for numbers lying on it, logarithmic identities can be used without fear, 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 main branch of the logarithmic function is continuous and differentiable on the entire complex plane, except for the negative part of the real axis, on which the imaginary part changes abruptly to $2\backslash pi$. But this fact is a consequence of the artificial limitation of the imaginary part of the main value by the interval $(-\backslash pi,\; \backslash pi]$. If we consider all branches of the function, then continuity occurs at all points except zero, where the function is not defined. If you resolve the curve $\backslash Gamma$ cross the negative part of the real axis, then the first such intersection transfers the result from the main value branch to the adjacent branch, and each subsequent intersection causes a similar shift along the branches of the logarithmic function (see figure).

From the analytic continuation formula it follows that on any branch of the logarithm:

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

For any circle $S$, covering the point $0$:

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

The integral is taken in the positive direction (counterclockwise). This identity underlies the theory of residues.

One can also define the analytic continuation of the complex logarithm using series known for the real case:

{{{2}}}

(Row 1)

{{{2}}}

(Row 2)

However, from the form of these series it follows that at one the sum of the series is equal to zero, that is, the series relates only to the main branch of the multivalued function of the complex logarithm. The radius of convergence of both series is 1.

Connection with inverse trigonometric and hyperbolic functions

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

Historical sketch

The first attempts to extend logarithms to complex numbers were made at the turn of the 17th-18th centuries by Leibniz and Johann Bernoulli, but they failed to create a holistic theory, primarily because the very concept of logarithm was not yet clearly defined. The discussion on this issue took place first between Leibniz and Bernoulli, and in the middle of the 18th century between D’Alembert and Euler. Bernoulli and D'Alembert believed that it should be determined $\backslash log(-x)\; =\; \backslash log(x)$, while Leibniz proved that the logarithm of a negative number is an imaginary number. The complete theory of logarithms of negative and complex numbers was published by Euler in 1747-1751 and is essentially no different from the modern one. Although the debate continued (D'Alembert defended his point of view and argued it in detail in an article in his Encyclopedia and in other works), Euler's approach received universal recognition by the end of the 18th century.

Write a review about the article "Complex logarithm"

Literature

Theory of logarithms

• Korn G., Korn T.. - M.: Nauka, 1973. - 720 p.
• Sveshnikov A. G., Tikhonov A. N. Theory of functions of a complex variable. - M.: Nauka, 1967. - 304 p.
• Fikhtengolts G. M. Course of differential and integral calculus. - ed. 6th. - M.: Nauka, 1966. - 680 p.

History of logarithms

• Mathematics of the 18th century // / Edited by A. P. Yushkevich, in three volumes. - M.: Science, 1972. - T. III.
• Kolmogorov A. N., Yushkevich A. P. (eds.). Mathematics of the 19th century. Geometry. Theory of analytic functions. - M.: Science, 1981. - T. II.

Notes

1. Logarithmic function. // . - M.: Soviet Encyclopedia, 1982. - T. 3.
2. , Volume II, pp. 520-522..
3. , With. 623..
4. , With. 92-94..
5. , With. 45-46, 99-100..
6. Boltyansky V. G., Efremovich V. A.. - M.: Nauka, 1982. - P. 112. - (Kvant Library, issue 21).
7. , Volume II, pp. 522-526..
8. , With. 624..
9. , With. 325-328..
10. Rybnikov K. A. History of mathematics. In two volumes. - M.: Publishing house. Moscow State University, 1963. - T. II. - P. 27, 230-231..
11. , With. 122-123..
12. Klein F.. - M.: Science, 1987. - T. II. Geometry. - pp. 159-161. - 416 s.

An excerpt characterizing the Complex logarithm

It was clear that this strong, strange man was under the irresistible influence exerted on him by this dark, graceful, loving girl.
Rostov noticed something new between Dolokhov and Sonya; but he did not define to himself what kind of new relationship this was. “They are all in love with someone there,” he thought about Sonya and Natasha. But he was not as comfortable with Sonya and Dolokhov as before, and he began to be at home less often.
Since the autumn of 1806, everything again started talking about the war with Napoleon even more fervently than last year. Not only were recruits appointed, but also 9 more warriors out of a thousand. Everywhere they cursed Bonaparte with anathema, and in Moscow there was only talk about the upcoming war. For the Rostov family, the whole interest of these preparations for war lay only in the fact that Nikolushka would never agree to stay in Moscow and was only waiting for the end of Denisov’s leave in order to go with him to the regiment after the holidays. The upcoming departure not only did not prevent him from having fun, but also encouraged him to do so. He spent most of his time outside the house, at dinners, evenings and balls.

XI
On the third day of Christmas, Nikolai had dinner at home, which Lately rarely happened to him. It was officially a farewell dinner, since he and Denisov were leaving for the regiment after Epiphany. About twenty people were having lunch, including Dolokhov and Denisov.
Never in the Rostov house did the air of love, the atmosphere of love, make itself felt with such force as on these holidays. “Catch moments of happiness, force yourself to love, fall in love yourself! Only this one thing is real in the world - the rest is all nonsense. And that’s all we’re doing here,” said the atmosphere. Nikolai, as always, having tortured two pairs of horses and not having had time to visit all the places where he needed to be and where he was called, arrived home just before lunch. As soon as he entered, he noticed and felt the tense, loving atmosphere in the house, but he also noticed a strange confusion reigning between some of the members of the society. Sonya, Dolokhov, the old countess and a little Natasha were especially excited. Nikolai realized that something was going to happen before dinner between Sonya and Dolokhov, and with his characteristic sensitivity of heart he was very gentle and careful during dinner in dealing with both of them. On the same evening of the third day of the holidays there was to be one of those balls at Yogel (the dance teacher), which he gave on holidays for all his students and female students.
- Nikolenka, will you go to Yogel? Please go,” Natasha told him, “he especially asked you, and Vasily Dmitrich (it was Denisov) is going.”
“Wherever I go on the orders of Mr. Athena!” said Denisov, who jokingly placed himself in the Rostov house on the foot of the knight Natasha, “pas de chale [dance with a shawl] is ready to dance.”
- If I have time! “I promised the Arkharovs, it’s their evening,” Nikolai said.
“And you?...” he turned to Dolokhov. And just now I asked this, I noticed that this shouldn’t have been asked.
“Yes, maybe...” Dolokhov answered coldly and angrily, looking at Sonya and, frowning, with exactly the same look as he looked at Pierre at the club dinner, he looked again at Nikolai.
“There is something,” thought Nikolai, and this assumption was further confirmed by the fact that Dolokhov left immediately after dinner. He called Natasha and asked what was it?
“I was looking for you,” Natasha said, running out to him. “I told you, you still didn’t want to believe,” she said triumphantly, “he proposed to Sonya.”
No matter how little Nikolai did with Sonya during this time, something seemed to come off in him when he heard this. Dolokhov was a decent and in some respects a brilliant match for the dowry-free orphan Sonya. From the point of view of the old countess and the world, it was impossible to refuse him. And therefore Nikolai’s first feeling when he heard this was anger against Sonya. He was preparing to say: “And great, of course, we must forget our childhood promises and accept the offer”; but he didn’t have time to say it yet...
– You can imagine! She refused, completely refused! – Natasha spoke. “She said she loves someone else,” she added after a short silence.
“Yes, my Sonya could not have done otherwise!” thought Nikolai.
“No matter how much my mother asked her, she refused, and I know she won’t change what she said...
- And mom asked her! – Nikolai said reproachfully.
“Yes,” said Natasha. - You know, Nikolenka, don’t be angry; but I know that you will not marry her. I know, God knows why, I know for sure, you won’t get married.
“Well, you don’t know that,” said Nikolai; – but I need to talk to her. What a beauty this Sonya is! – he added smiling.
- This is so lovely! I'll send it to you. - And Natasha, kissing her brother, ran away.
A minute later Sonya came in, frightened, confused and guilty. Nikolai approached her and kissed her hand. This was the first time on this visit that they spoke face to face and about their love.
“Sophie,” he said timidly at first, and then more and more boldly, “if you want to refuse not only a brilliant, profitable match; but he is a wonderful, noble man... he is my friend...
Sonya interrupted him.
“I already refused,” she said hastily.
- If you refuse for me, then I’m afraid that on me...
Sonya interrupted him again. She looked at him with pleading, frightened eyes.
“Nicolas, don’t tell me that,” she said.
- No, I have to. Maybe this is suffisance [arrogance] on my part, but it’s better to say. If you refuse for me, then I must tell you the whole truth. I love you, I think, more than anyone...
“That’s enough for me,” Sonya said, flushing.
- No, but I have fallen in love a thousand times and will continue to fall in love, although I do not have such a feeling of friendship, trust, love for anyone as for you. Then I'm young. Maman doesn't want this. Well, it's just that I don't promise anything. And I ask you to think about Dolokhov’s proposal,” he said, having difficulty pronouncing his friend’s last name.
- Don't tell me that. I do not want anything. I love you like a brother, and will always love you, and I don’t need anything more.
“You are an angel, I am not worthy of you, but I am only afraid of deceiving you.” – Nikolai kissed her hand again.

Yogel had the most fun balls in Moscow. This was what the mothers said, looking at their adolescentes [girls] performing their newly learned steps; this was said by the adolescentes and adolescents themselves, [girls and boys] who danced until they dropped; these grown-up girls and young men who came to these balls with the idea of ​​condescending to them and finding the best fun in them. In the same year, two marriages took place at these balls. The two pretty princesses of the Gorchakovs found suitors and got married, and even more so they launched these balls into glory. What was special about these balls was that there was no host and hostess: there was the good-natured Yogel, like flying feathers, shuffling around according to the rules of art, who accepted tickets for lessons from all his guests; It was that only those who wanted to dance and have fun, like 13 and 14 year old girls who put on long dresses for the first time, want to go to these balls. Everyone, with rare exceptions, was or seemed pretty: they all smiled so enthusiastically and their eyes lit up so much. Sometimes even the best students danced pas de chale, of whom the best was Natasha, distinguished by her grace; but at this last ball only ecosaises, anglaises and the mazurka, which was just coming into fashion, were danced. The hall was taken by Yogel to Bezukhov’s house, and the ball was a great success, as everyone said. There were a lot of pretty girls, and the Rostov ladies were among the best. They were both especially happy and cheerful. That evening, Sonya, proud of Dolokhov’s proposal, her refusal and explanation with Nikolai, was still spinning at home, not allowing the girl to finish her braids, and now she was glowing through and through with impetuous joy.
Natasha, no less proud that she was in long dress, at the real ball, she was even happier. Both were wearing white muslin dresses with pink ribbons.
Natasha became in love from the very minute she entered the ball. She was not in love with anyone in particular, but she was in love with everyone. The one she looked at at the moment she looked at was the one she was in love with.
- Oh, how good! – she kept saying, running up to Sonya.
Nikolai and Denisov walked around the halls, looking at the dancers affectionately and patronizingly.
“How sweet she will be,” Denisov said.
- Who?
“Athena Natasha,” answered Denisov.
“And how she dances, what a g”ation!” after a short silence, he said again.
- Who are you talking about?
“About your sister,” Denisov shouted angrily.
Rostov grinned.
– Mon cher comte; vous etes l"un de mes meilleurs ecoliers, il faut que vous dansiez,” said little Jogel, approaching Nikolai. “Voyez combien de jolies demoiselles.” [My dear Count, you are one of my best students. You need to dance. Look how much pretty girls!] – He made the same request to Denisov, also his former student.
“Non, mon cher, je fe"ai tapisse"ie, [No, my dear, I’ll sit by the wall," Denisov said. “Don’t you remember how badly I used your lessons?”
- Oh no! – Jogel said hastily consoling him. – You were just inattentive, but you had abilities, yes, you had abilities.
The newly introduced mazurka was played; Nikolai could not refuse Yogel and invited Sonya. Denisov sat down next to the old ladies and, leaning his elbows on his saber, stamping his beat, told something cheerfully and made the old ladies laugh, looking at the dancing young people. Yogel, in the first couple, danced with Natasha, his pride and best student. Gently, tenderly moving his feet in his shoes, Yogel was the first to fly across the hall with Natasha, who was timid, but diligently performing steps. Denisov did not take his eyes off her and tapped the beat with his saber, with an expression that clearly said that he himself did not dance only because he did not want to, and not because he could not. In the middle of the figure, he called Rostov, who was passing by, to him.
“It’s not the same at all,” he said. - Is this a Polish mazurka? And she dances excellently. - Knowing that Denisov was even famous in Poland for his skill in dancing the Polish mazurka, Nikolai ran up to Natasha:
- Go and choose Denisov. Here he is dancing! Miracle! - he said.
When Natasha’s turn came again, she stood up and quickly fingering her shoes with bows, timidly, ran alone across the hall to the corner where Denisov was sitting. She saw that everyone was looking at her and waiting. Nikolai saw that Denisov and Natasha were arguing smiling, and that Denisov was refusing, but smiling joyfully. He ran up.
“Please, Vasily Dmitrich,” Natasha said, “let’s go, please.”
“Yes, that’s it, g’athena,” Denisov said.
“Well, that’s enough, Vasya,” said Nikolai.
“It’s like they’re trying to persuade Vaska the cat,” Denisov said jokingly.
“I’ll sing to you all evening,” said Natasha.
- The sorceress will do anything to me! - Denisov said and unfastened his saber. He came out from behind the chairs, firmly took his lady by the hand, raised his head and put his foot down, waiting for tact. Only on horseback and in the mazurka it was not visible vertically challenged Denisov, and he seemed to be the same young man that he felt himself to be. Having waited for the beat, he glanced triumphantly and playfully at his lady from the side, suddenly tapped one foot and, like a ball, elastically bounced off the floor and flew along in a circle, dragging his lady with him. He silently flew halfway across the hall on one leg, and it seemed that he did not see the chairs standing in front of him and rushed straight towards them; but suddenly, clicking his spurs and spreading his legs, he stopped on his heels, stood there for a second, with the roar of spurs, knocked his feet in one place, quickly turned around and, clicking his right foot with his left foot, again flew in a circle. Natasha guessed what he intended to do, and, without knowing how, she followed him - surrendering herself to him. Now he circled her, now on his right, now on his left hand, now falling on his knees, he circled her around himself, and again he jumped up and ran forward with such swiftness, as if he intended to run across all the rooms without taking a breath; then suddenly he stopped again and again made a new and unexpected knee. When he, briskly spinning the lady in front of her place, snapped his spur, bowing before her, Natasha did not even curtsey for him. She stared at him in bewilderment, smiling as if she didn’t recognize him. - What is this? - she said.
Despite the fact that Yogel did not recognize this mazurka as real, everyone was delighted with Denisov’s skill, they began to choose him incessantly, and the old people, smiling, began to talk about Poland and the good old days. Denisov, flushed from the mazurka and wiping himself with a handkerchief, sat down next to Natasha and did not leave her side throughout the entire ball.

Logarithmic function

A logarithmic function is a function of the form f(x) = logax, defined at

Domain: . Range of values: . The function is strictly increasing for a > 1 and strictly decreasing for 0< a < 1. График любой логарифмической функции проходит через точку (1;0). Функция непрерывна и неограниченно дифференцируема всюду в своей области определения.

The straight line x = 0 is a left vertical asymptote, since for a > 1 and for 0< a < 1.

The derivative of the logarithmic function is equal to:

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

Complex logarithm

Definition and properties

For complex numbers, the logarithm is defined in the same way as a real one. In practice, the natural complex logarithm is used almost exclusively, which we denote and define as the set of all complex numbers z such that ez = w. The complex logarithm exists for anyone, and its real part is determined uniquely, while the imaginary part has infinite set values. For this reason it is called a multi-valued function. If we represent w in exponential form:

then the logarithm is found by the formula:

Here is a real logarithm, r = | w | , k is an arbitrary integer. The value obtained when k = 0 is called the principal value of the complex natural logarithm; It is customary to take the value of the argument in the interval (? р,р]. The corresponding (already single-valued) function is called the main branch of the logarithm and is denoted. Sometimes the value of the logarithm that does not lie on the main branch is also denoted by.

From the formula it follows:

The real part of the logarithm is determined by the formula:

The logarithm of a negative number is found using the formula.

The exponential function of a real variable (with a positive base) is determined in several steps. First, for natural values ​​- as a product of equal factors. The definition then extends to negative integers and non-zero values ​​for by the rules. Next, we consider fractional indicators at which the value exponential function determined using roots: . For irrational values, the definition is already connected with the basic concept of mathematical analysis - with the passage to the limit, for reasons of continuity. All these considerations are in no way applicable to attempts to extend the exponential function to complex values ​​of the indicator, and what it is, for example, is completely unclear.

For the first time, a power with a complex exponent with a natural base was introduced by Euler based on an analysis of a number of constructions of integral calculus. Sometimes very similar algebraic expressions, when integrated, give completely different answers:

At the same time, here the second integral is formally obtained from the first when replaced by

From this we can conclude that with the proper definition of an exponential function with a complex exponent, inverse trigonometric functions are related to logarithms and thus the exponential function is related to trigonometric ones.

Euler had the courage and imagination to give a reasonable definition for an exponential function with a base, namely,

This is a definition, and therefore this formula cannot be proven; one can only look for arguments in favor of the reasonableness and expediency of such a definition. Mathematical analysis provides a lot of arguments of this kind. We will limit ourselves to just one.

It is known that for real there is a limiting relation: . On the right side there is a polynomial that also makes sense for complex values ​​for . The limit of a sequence of complex numbers is determined naturally. A sequence is considered convergent if the sequences of real and imaginary parts converge and is accepted

Let's find it. To do this, let's turn to the trigonometric form and for the argument we will select values ​​from the interval. With this choice it is clear that for . Further,

To go to the limit, you need to verify the existence of limits for and and find these limits. It is clear that

So, in the expression

the real part tends to , the imaginary part tends to so

This simple argument provides one of the arguments in favor of Euler's definition of the exponential function.

Let us now establish that when multiplying the values ​​of an exponential function, the exponents add up. Really:

2. Euler's formulas.

Let us put in the definition of the exponential function . We get:

Replacing b with -b, we get

By adding and subtracting these equalities term by term, we find the formulas

called Euler's formulas. They establish a connection between trigonometric functions and exponential with imaginary exponents.

3. Natural logarithm of a complex number.

A complex number given in trigonometric form can be written in the form. This form of writing a complex number is called exponential. It retains all the good properties of trigonometric form, but is even more concise. Further, Therefore, it is natural to assume that the real part of the logarithm of a complex number is the logarithm of its modulus, and the imaginary part is its argument. This to some extent explains the “logarithmic” property of the argument - the argument of the product is equal to the sum of the arguments of the factors.

The exponential function of a real variable (with a positive base) is determined in several steps. First, for natural values ​​- as a product of equal factors. The definition then extends to negative integers and non-zero values ​​for by the rules. Next, we consider fractional exponents in which the value of the exponential function is determined using roots: . For irrational values, the definition is already connected with the basic concept of mathematical analysis - with the passage to the limit, for reasons of continuity. All these considerations are in no way applicable to attempts to extend the exponential function to complex values ​​of the indicator, and what it is, for example, is completely unclear.

For the first time, a power with a complex exponent with a natural base was introduced by Euler based on an analysis of a number of constructions of integral calculus. Sometimes very similar algebraic expressions, when integrated, give completely different answers:

At the same time, here the second integral is formally obtained from the first when replaced by

From this we can conclude that with the proper definition of an exponential function with a complex exponent, inverse trigonometric functions are related to logarithms and thus the exponential function is related to trigonometric ones.

Euler had the courage and imagination to give a reasonable definition for an exponential function with a base, namely,

This is a definition, and therefore this formula cannot be proven; one can only look for arguments in favor of the reasonableness and expediency of such a definition. Mathematical analysis provides many arguments of this kind. We will limit ourselves to just one.

It is known that for real there is a limiting relation: . On the right side there is a polynomial that also makes sense for complex values ​​for . The limit of a sequence of complex numbers is determined naturally. A sequence is considered convergent if the sequences of real and imaginary parts converge and is accepted

Let's find it. To do this, let's turn to the trigonometric form and for the argument we will select values ​​from the interval. With this choice it is clear that for . Further,

To go to the limit, you need to verify the existence of limits for and and find these limits. It is clear that

So, in the expression

the real part tends to , the imaginary part tends to so

This simple argument provides one of the arguments in favor of Euler's definition of the exponential function.

Let us now establish that when multiplying the values ​​of an exponential function, the exponents add up. Really:

2. Euler's formulas.

Let us put in the definition of the exponential function . We get:

Replacing b with -b, we get

By adding and subtracting these equalities term by term, we find the formulas

called Euler's formulas. They establish a connection between trigonometric functions and exponential functions with imaginary exponents.

3. Natural logarithm of a complex number.

A complex number given in trigonometric form can be written in the form. This form of writing a complex number is called exponential. It retains all the good properties of trigonometric form, but is even more concise. Further, Therefore, it is natural to assume that the real part of the logarithm of a complex number is the logarithm of its modulus, and the imaginary part is its argument. This to some extent explains the “logarithmic” property of the argument - the argument of the product is equal to the sum of the arguments of the factors.