What is Complex analysis: Definition and 778 Discussions

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.As a differentiable function of a complex variable is equal to its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions).

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  1. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Proof of the Arzela-Ascoli Theorem for Functions

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Proof of the Arzela-Ascoli Theorem for Functions

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  2. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Proof of the Arzela-Ascoli Theorem for Functions

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Proof of the Arzela-Ascoli Theorem for Functions

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  3. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Introduction to the Montel Theorem

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Introduction to the Montel Theorem

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  4. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Completion of Proof of the Montel Theorem

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Completion of Proof of the Montel Theorem

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  5. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Introduction to Marty's Theorem

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Introduction to Marty's Theorem

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  6. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Proof of one direction of Marty's Theorem

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Proof of one direction of Marty's Theorem

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  7. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Proof of the other direction of Marty's Theorem

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Proof of the other direction of Marty's Theorem

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  8. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Normal Convergence at Infinity and Hurwitz's Theorems

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Normal Convergence at Infinity and Hurwitz's Theorems

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  9. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Normal Sequential Compactness, Normal Uniform Boundedness

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Normal Sequential Compactness, Normal Uniform Boundedness

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  10. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Local Analysis of Normality and the Zooming Process - Motivation for Zalcman's Lemma

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Local Analysis of Normality and the Zooming Process - Motivation for Zalcman's Lemma

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  11. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Characterizing Normality at a Point by the Zooming Process

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Characterizing Normality at a Point by the Zooming Process

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  12. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Local Analysis of Normality and the Zooming Process - Motivation for Zalcman's Lemma

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Local Analysis of Normality and the Zooming Process - Motivation for Zalcman's Lemma

    COPYRIGHT strictly reserved to Dr. T.E. Venkata Balaji (IIT Madras) and NPTEL, Govt of India. Duplication PROHIBITED.Lectures: http://www.nptel.ac.in/courses/111106094/Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111106094
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  13. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Montel's Deep Theorem: The Fundamental Criterion for Normality

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Montel's Deep Theorem: The Fundamental Criterion for Normality

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  14. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Proofs of the Great and Little Picard Theorems

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Proofs of the Great and Little Picard Theorems

    COPYRIGHT strictly reserved to Dr. T.E. Venkata Balaji (IIT Madras) and NPTEL, Govt of India. Duplication PROHIBITED.Lectures: http://www.nptel.ac.in/courses/111106094/Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111106094
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  15. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Royden\'s Theorem on Normality Based On Growth Of Derivatives

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Royden\'s Theorem on Normality Based On Growth Of Derivatives

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  16. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Schottky's Theorem: Uniform Boundedness from a Point to a Neighbourhood

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Schottky's Theorem: Uniform Boundedness from a Point to a Neighbourhood

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  17. mjtsquared

    I Region of convergence of a Laplace transform

    If a Laplace transform has a region of convergence starting at Re(s)=0, does the Laplace transform evaluated at the imaginary axis exist? I.e. say that the Laplace transform of 1 is 1/s. Does this Laplace transform exist at say s=i?
  18. R

    What is the Taylor expansion of x/sin(ax)?

    Hey everyone 1. Homework Statement I want to compute the Taylor expansion (the first four terms) of $$f(x) =x/sin(ax)$$ around $$x_0 = 0$$. I am working in the space of complex numbers here. Homework Equations function: $$f(x) = \frac{x}{\sin (ax)}$$ Taylor expansion: $$ f(x) = \sum...
  19. T

    Complex Analysis prerequisite material review

    Homework Statement Identify the set of points satisfying ##1<\vert 2z-6\vert <2## such that ##z\in\Bbb{C}##. My pre-caculus is very rusty, so I am not sure if I am doing this correctly. Homework Equations ##x^2 +y^2= r^2## ##\forall z,z'\in\Bbb{C}, \vert zz'\vert =\vert z\vert\vert z'\vert##...
  20. Rectifier

    Complex logarithm as primitive

    The problem I am trying to calculate the integral $$ \int_{\gamma} \frac{z}{z^2+4} \ dz $$ Where ## \gamma ## is the line segment from ## z=2+2i ## to ## z=-2-2i ##. The attempt I would like to solve this problem using substitution and a primitive function to ## \frac{1}{u} ##. I am not...
  21. Rectifier

    Solving Complex Equation: $$ \bar{z} = z^n $$

    The problem I would like to solve: $$ \bar{z} = z^n $$ where ##n## is a positive integer. The attempt ## z = r e^{i \theta} \\ \\ \overline{ r e^{i \theta} } = r^n e^{i \theta n} \\ r e^{-i \theta} = r^n e^{i \theta n} ## ## r = r^n \Leftrightarrow true \ \ if \ \ n=1 \ \ or \ \ r=1## ##...
  22. L

    MHB [Complex Analysis] Singularity in product of analytic functions

    Suppose f,g:ℂ→ℂ are analytic with singularities at z=0. I was wondering whether f(z)^2 or f(z)g(z) will have a singularity at z=0? For each, can you give me a proof or a counterexample?
  23. Peter Alexander

    Understanding the Cauchy Integration Formula for Analytic Functions

    Hello everyone! I'm having a bit of a problem with comprehension of the Cauchy integration formula. I might be missing some key know-how, so I'm asking for any sort of help and/or guideline on how to tackle similar problems. I thank anyone willing to take a look at my post! Homework Statement...
  24. A

    Laurent series of z^2sin(1/(z-1))

    Homework Statement Find Laurent series of $$z^2sin(\frac{1}{1-z})$$ at $$0<\lvert z-1 \rvert<\infty$$ Homework Equations sine series expansion. The Attempt at a Solution At first, it seems pretty elementary since you can set w=\frac{1}{z-1} and expand at infinity in z, which is 0 in w...
  25. C

    How Do You Solve a Cubic Equation Using Cardano's Formula?

    Homework Statement Use Cardano's formula to find a real root for ##3x^3-45x^2+243x-525=0##. [Edited to correct mistake] Homework Equations $$x = u - \frac{b}{3a}$$ Depressed cubic: $$u^3=3pu+2q$$ Cardano's formula: $$u=\sqrt[3]{q+\sqrt{q^2-p^3}}+\sqrt[3]{q-\sqrt{q^2-p^3}}$$ The Attempt at a...
  26. Jamz

    I Map from space spanned by 2 complex conjugate vars to R^2

    Hello, I would like your help understanding how to map a region of the space \mathbb{C}^2 spanned by two complex conjugate variables to the real plane \mathbb{R}^2 . Specifically, let us think that we have two complex conugate variables z and \bar{ z} and we define a triangle in the...
  27. Math Amateur

    MHB Smooth Paths in Complex Analysis .... Palka Example 1.3, Section 1.2 in Chapter 4 .... ....

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter 4: Complex Integration, Section 1.2 Smooth and Piecewise Smooth Paths ... I need help with some aspects of Example 1.3, Section 1.2, Chapter 4 ... Example 1.3, Section 1.2, Chapter 4...
  28. M

    Help with these two problems in complex analysis

    Homework Statement What is the argument of -4-3i, and -4+3i? Homework Equations tantheta=opposite/adjacent side The principle of argument is that the argument lies between -pi and pi (not including -pi). The Attempt at a Solution arg(-4-3i) = -pi + arctan(3/4) arg(-4+3i) = pi - arctan(3/4)...
  29. T

    I What does ".... is zero to order ...." mean?

    I saw the sentence " So the contour integral of an analytic function f(z) around a tiny square of size e is zero to order e^2. ". I want to know what " be zero to order " means exactly.
  30. L

    I Can a Circular Function with Complex Variable Represent a 3D Graph?

    Does a circular function with complex variable represent a three-dimensional graph? For example cosiz
  31. J

    I Help with expression ##F(it)-F(-it)## in the Abel-Plana form

    I´m having a problem with the value of the expression ##F(it)-F(-it)##, found on the Abel-Plana formula, where $$F(z)=\sqrt{z^2 + A^2}$$, with ##A## being a positive real number (F(z) is analytic in the right half-plane). Well, I know the result is ##F(it)-F(-it)=2i\sqrt{t^2 -A^2}##, for...
  32. Math Amateur

    MHB What Is the Top Recommended Book on Complex Analysis for Beginners on MHB?

    What book do MHB members regard as the best for a rigorous but clear and (moderately) easily understood introduction to complex analysis? (Note - would be good if the book had hints to solutions of exercise.) Peter
  33. C

    A Question about derivatives of complex fields

    https://arxiv.org/pdf/1705.07188.pdf Equation 5 in this paper states that $$\frac{\partial F}{\partial p_i} = 2Re\left\lbrace\frac{\partial F}{\partial x}\frac{\partial x}{\partial p_i}\right\rbrace$$ Here, p_i stands for the i'th element of a vector of 'design parameters' \mathbf{p}. These...
  34. C

    Analysis Are there any recommended Complex Analysis books for advanced students?

    I'm looking for a good Complex book, but the options seem slim. I was thinking about Rudin's Real and Complex. My only reservation is that it is not structured like any other book I've seen. I've had advanced analysis and measure and integration theory, so rigour is not a concern. I saw Alfohr's...
  35. S

    I Proof of Harmonic Function Infinitely Differentiable

    Hello! I have this Proposition: "A harmonic function is infinitely differentiable". The book gives a proof that uses this theorem: "Suppose u is harmonic on a simply-connected region G. Then there exists a harmonic function v in G such that ##f = u + iv## is holomorphic in G. ". In the proof...
  36. S

    Proof of Degree <= 1 for Entire Function f

    Homework Statement Suppose f is entire and there exist constants a and b such that ##|f(z)| \le a|z|+b## for all ##z \in C##. Prove that f is a polynomial of degree at most 1. Homework EquationsThe Attempt at a Solution We have that for any ##z \neq 0##, ##\frac{|f(z)|}{a|z|} \le b##. So if we...
  37. T

    Another Improper Integral Using Complex Analysis

    Homework Statement $$\int_{-\infty}^\infty \space \frac{cos(2x)}{x-3i}dx$$ Homework EquationsThe Attempt at a Solution $$\int_{-R}^R \space \frac{e^{2ix}}{x-3i}dx + \int_{C_R} \space \frac{e^{2iz}}{z-3i}dz = 2\pi i\sum\space res \space f(z)$$ Then using Jordan's Lemma, as ##R\to\infty## the...
  38. T

    Improper Integral Using Complex Analysis

    Homework Statement Compute the Integral: ##\int_{-\infty}^\infty \space \frac{e^{-2ix}}{x^2+4}dx## Homework Equations ##\int_C \space f(z) = 2\pi i \sum \space res \space f(z)## The Attempt at a Solution At first I tried doing this using a bounded integral but couldn't seem to get the right...
  39. K

    Complex Analysis/Radius of Convergence question.

    Homework Statement Question asks to show that if f is an entire function and bounded then it is polynomial of degree m or less. Homework Equations The Attempt at a Solution I tried plugging in the power series for f(z) and tried/know it is related to Liouville's Theorem somehow but I am...
  40. K

    Complex Analysis. Laurent Series Expansion in region(22C).

    <Moderator's note: moved from a technical forum, so homework template missing> Hi. I have solved the others but I am really struggling on 22c. I need it to converge for |z|>2. This is the part I am really struggling with. I am trying to get both fractions into a geometric series with...
  41. ZuperPosition

    Possible branch cuts for arcsin derivative

    Homework Statement Our textbook, Fundamentals of Complex Analysis, (...) by Saff Snider says on page 135 that by choosing some suitable branch for the square root and the logarithm then one can show that any such branch satisfies the equation below. The homework/task is to find all such branch...
  42. F

    I Deformation of contour of integration or shifting poles

    As I understand it, in order to compute a contour integral one can deform the contour of integration, such that it doesn't pass through any poles of the integrand, and the result is identical to that found using the original contour of integration considered. However, I have seen applications...
  43. Alettix

    Antiderivative of 1/x: ln(x) or ln(|x|)?

    Homework Statement Calculate the integral: ## \int_{a}^{b} \frac{1}{x} dx ## Homework Equations - The Attempt at a Solution In high school we learned that: ## \int_{a}^{b} \frac{1}{x} dx = ln(|x|) + C ## because the logarithm of a negative number is undefined. However, in my current maths...
  44. V

    A Relativity in Complex Analysis: Is There a Formulation?

    Is there a formulation of any of the relativity theories in terms of complex analysis? As in - I imagine - every event would be a complex number in a complex field.. or something as such..
  45. arpon

    Complex Integration using residue theorem

    Homework Statement [/B] ##C_\rho## is a semicircle of radius ##\rho## in the upper-half plane. What is $$\lim_{\rho\rightarrow 0} \int_{C_{\rho}} \frac{e^{iaz}-e^{ibz}}{z^2} \,dz$$Homework Equations If ##C## is a closed loop and ##z_1, z_2 ... z_n## are the singular points inside ##C##...
  46. J

    I Can the Complex Integral Problem Be Solved Using Residue Theorem?

    I have this problem with a complex integral and I'm having a lot of difficulty solving it: Show that for R and U both greater than 2a, \exists C > 0, independent of R,U,k and a, such that $$\int_{L_{-R,U}\cup L_{R,U}} \lvert f(z)\rvert\,\lvert dz\rvert \leqslant \frac{C}{kR}.$$ Where a > 0, k...
  47. J

    I Problem with this estimation lemma example

    I have been trying to show that $$\lim_{U\rightarrow\infty}\int_C \frac{ze^{ikz}}{z^2+a^2}dz = 0 $$ Where $$R>2a$$ and $$k>0$$ And C is the curve, defined by $$C = {x+iU | -R\le x\le R}$$ I have tried by using the fact that $$|\int_C \frac{ze^{ikz}}{z^2+a^2}dz| \le\int_C...
  48. G

    A A problem about branch cut in contour integral

    Hello. I have a difficulty to understand the branch cut introduced to solve this integral. \int_{ - \infty }^\infty {dp\left[ {p{e^{ip\left| x \right|}}{e^{ - it\sqrt {{p^2} + {m^2}} }}} \right]} here p is a magnitude of the 3-dimensional momentum of a particle, x and t are space and time...
  49. dykuma

    Contour integral using residue theorem

    Homework Statement Find the solution of the following integral Homework Equations The Attempt at a Solution I applied the above relations getting that Then I was able to factor the function inside the integral getting that From here I should be able to get a solution by simply finding the...
  50. G

    I Domain of single-valued logarithm of complex number z

    Hello. Let's have any non-zero complex number z = reiθ (r > 0) and natural log ln applies to z. ln(z) = ln(r) + iθ. In fact, there is an infinite number of values of θ satistying z = reiθ such as θ = Θ + 2πn where n is any integer and Θ is the value of θ satisfying z = reiθ in a domain of -π <...
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