Complex analysis Definition and 180 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. Proving that a "composition" is harmonic

I am learning some complex analysis as it is a prerequisite for the masters program that I was accepted into and I didn't take it yet during my bachelors. I am using some lecture notes in Slovene and I have run into a problem that has proven troublesome for me : If ##g: D \rightarrow \mathbb{C}...
2. Classifying singularities of a function

I came across this question on chegg for practice as I'm self learning complex analysis, but became stumped on it and without access to the solution am unable to check. Let $$f(z)=\frac{cos(z)} {(z-π/2)^7}$$. Then the singularity is at π/2. And on first appearance, it looks like a pole of...
3. I Sum of the dot product of complex vectors

Summary:: summation of the components of a complex vector Hi, In my textbook I have ##\widetilde{\vec{E_t}} = (\widetilde{\vec{E_i}} \cdot \hat{e_p}) \hat{e_p}## ##\widetilde{\vec{E_t}} = \sum_j( (\widetilde{\vec{E_{ij}}} \cdot {e_{p_j}}*) \hat{e_p}## For ##\hat{e_p} = \hat{x}##...
4. Proving geometric sum for complex numbers

I went ahead and tried to prove by induction but I got stuck at the base case for ## N =1 ## ( in my course we don't define ## 0 ## as natural so that's why I started from ## N = 1 ## ) which gives ## \sum_{k=0}^1 z_k = 1 + z = 1+ a + ib ## . I need to show that this is equal to ## \frac{1-...
5. Engineering Book considering FEM analysis for complex eigenvalues (incl. damping)

Can anyone recommend a book in which complex eigenvalue problems are treated? I mean the FEM analysis and the theory behind it. These are eigenvalue problems which include damping. I think that it is used for composite materials and/or airplane engineering (maybe wing fluttering?).

11. [Complex analysis] Contradiction in the definition of a branch

I find the following definition in my complex analysis book : Definition : ## F(z)## is said to be a branch of a multiple-valued function ##f(z)## in a domain ##D## if ##F(z)## is single-valued and continuous in ##D## and has the property that, for each ##z## in ##D##, the value ##F(z)## is one...
12. Show that the real part of a certain complex function is harmonic

Hello, I have to prove that the complex valued function $$f(z) = Re\big(\frac{\cos z}{\exp{z}}\big)$$ is harmonic on the whole complex plane. This exercice immediately follows a chapter on the extension of the usual functions (trigonometric and the exponential) to the complex plane, so I tend...
13. How to simplify this complex expression?

I don't know how to start with the factorization. $$\frac{(-1)^{2/9} + (-3/2 - \frac{i}{2} \sqrt{3})^{(1/3)}}{(-1)^{2/9}- (-3/2 - \frac{i}{2} \sqrt{3})^{(1/3)}}$$ Any hints would be nice. Thank you.
14. Laurent Series (Complex Analysis)

My homework is on mathematical physics and I want to know the concept behind Laurent series. I want to know clearly know the process behind attaining the series representation for the expansion in sigma notation using the formula that can be found on the attached files. There are three questions...

27. I Showing that a function is analytic

Say we have ##P_k(z)## a family of entire functions, and they depend analytically on ##k## in ##\Delta##. Assume ##P_k(z)## is nonzero on ##S^1## for all ##k##. How do I see that for each ##t \ge 0##, we have that$$\sum_{|z| < 1, P_k(z) = 0} z^t$$is an analytic function of ##k##? Here, the zeros...
28. How to write the complex exponential in terms of sine/cosine?

I apologize in advance if any formatting is weird; this is my first time posting. If I am breaking any rules with the formatting or if I am not providing enough detail or if I am in the wrong sub-forum, please let me know. 1. Homework Statement Using Euler's formula : ejx = cos(x) + jsin(x)...
29. Engineering Problems about Zin in complex circuit analysis

1. Homework Statement the problem is my answer for question (a) is not the same as the answer provided by the question, i get 2.81 - j4.49 Ω while the answer demands 2.81 + j4.49 Ω Homework Equations simplifying the circuit, details can be seen below The Attempt at a Solution...
30. Online app which plots F(z) in the complex plane

I am looking for an app that can instantaneously plot the function f(z) in the complex plane once z is given. It would be much favorable if this process is fast which allows one to visualize f(z) when the user is moving the mouse on the complex plane to the location of z. One possible...
31. A Order of poles of z∕Sin(πz²)

When The denominator is checked, the poles seem to be at Sin(πz²)=0, Which means πz²=nπ ⇒z=√n for (n=0,±1,±2...) but in the solution of this problem, it says that, for n=0 it would be simple pole since in the Laurent expansion of (z∕Sin(πz²)) about z=0 contains the highest negative power to be...
32. I Equating coefficients of complex exponentials

I have an equation that looks like ##i\dot{\psi_n}=X~\psi_n+\frac{C~\psi_n+D~a~\psi^\ast_{n+1}+E~b~\psi_{n+1}}{1+\beta~(D~\psi^\ast_{n+1}+E~\psi_{n+1})}## where ##E,b,D,a,C,X## are constants. I have the ansatz ##\psi_n=A_n~e^{ixt}+B^\ast_n~e^{-itx^\ast}##, ##x## and ##A_n,B_n## are complex...
33. Laurent expansion of ##ze^{1/z}##

Homework Statement Find a Laurent series of ##f(z)=ze^{1/z}## in powers of ##z-1##. Is there an easier way to go about this as this is not a typical expansion I see on textbooks. It seems that my incomplete solution is too complicated. Please help, exam is in two days and I am working on past...
34. Evaluating a complex integral

Homework Statement ##\int_{0}^{2\pi} cos^2(\frac{pi}{6}+2e^{i\theta})d\theta##. I am not sure if I am doing this write. Help me out. Thanks! Homework Equations Cauchy-Goursat's Theorem The Attempt at a Solution Let ##z(\theta)=2e^{i\theta}##, ##\theta \in [0,2\pi]##. Then the complex integral...
35. When is an entire function a constant?

Homework Statement Let ##f(z)## be an entire function of ##z \in \Bbb{C}##. If ##\operatorname{Im}(f(z)) \gt 0##, then ##f(z)## is a constant. Homework Equations n/a The Attempt at a Solution I don't get how the imaginary part of ##f(z)## would be greater than any number. Aren't complex...
36. Is this question incomplete? Regarding entire functions...

Homework Statement Let ##F## be an entire function such that ##\exists## positve constants ##c## and ##d## where ##\vert f(z)\vert \leq c+d\vert z\vert^n, \forall z\in \Bbb{C}##. Is this question incomplete? My complex analysis course is not rigorous at all and this came up on a past final...
37. Finding residues with Laurent series.

Homework Statement Use an appropriate Laurent series to find the indicated residue for ##f(z)=\frac{4z-6}{z(2-z)}## ; ##\operatorname{Res}(f(z),0)## Homework Equations n/a The Attempt at a Solution Computations are done such that ##0 \lt \vert z\vert \lt 2##...
38. Complex Analysis - sqrt(z^2 + 1) function behavior

Homework Statement Homework Equations The relevant equation is that sqrt(z) = e^(1/2 log z) and the principal branch is from (-pi, pi] The Attempt at a Solution The solution is provided, since this isn't a homework problem (I was told to post it here anyway). I don't understand why the...
39. I Confused by the behavior of sqrt(z^2+1)

(mentor note: this is a homework problem with a solution that the OP would like to understand better) In Taylor's Complex Variables, Example 1.4.10 Can someone help me understand this? I don't know what they mean by (i, i inf), or how they got it and -it
40. I Principal branch of the log function

I'm learning complex analysis right now, and I'm reading from Joseph Taylor's Complex Variables. On Theorem 1.4.8, it says "If a log is the branch of the log function determined by an interval I, then log agrees with the ordinary natural log function on the positive real numbers if and only if...
41. Contour Integration over Square, Complex Anaylsis

Homework Statement Show that $$\int_C e^zdz = 0$$ Let C be the perimeter of the square with vertices at the points z = 0, z = 1, z = 1 +i and z = i. Homework Equations $$z = x + iy$$ The Attempt at a Solution I know that if a function is analytic/holomorphic on a domain and the contour lies...
42. A What is the "real" Feynman propagator?

The logic of the Feynman Propagator is confusing to me. Written in integral form as it is below $$\Delta _ { F } ( x - y ) = \int \frac { d ^ { 4 } p } { ( 2 \pi ) ^ { 4 } } \frac { i } { p ^ { 2 } - m ^ { 2 } } e ^ { - i p \cdot ( x - y ) },$$ there are poles on the real axis. I have seen...
43. I Intuition - Cauchy integral theorem

So folks, I'm learning complex analysis right now and I've come across one thing that simply fails to enter my mind: the Cauchy Integral Theorem, or the Cauchy-Goursat Theorem. It says that, if a function is analytic in a certain (simply connected) domain, then the contour integral over a simple...
44. Where is ##(z+1)Ln(z)## differentiable?

Homework Statement Find the domain in which the complex-variable function ##f(z)=(z+1)Ln(z)## is differentiable. Note: ##Ln(z)## is the principal complex logarithmic function. Homework Equations Cuachy-Riemann Equations? The Attempt at a Solution The solution I have in mind would be to let...
45. I Verifying an equality

I have an expression ##\mathcal{Im}[RT^*e^{-2ip}]=|T|^2\sin p ##, where ##R=Ae^{ip}+Be^{-ip} ## and ##p ## is a real number. This ultimately should lead to ##\mathcal{Im}[A+B+Te^{2ip}]=0 ## upto a sign (perhaps if I didn't do a mistake). There is a condition on ##R ## that it is real...
46. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Properties of the Image of an Analytic Function: Introduction to the 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
47. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Recalling Singularities of Analytic Functions: Non-isolated and Isolated Removable

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
48. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Recalling Riemann's Theorem on Removable Singularities

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
49. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Casorati-Weierstrass Theorem; Dealing with the Point at Infinity

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
50. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Neighbourhood of Infinity, Limit at Infinity and Infinity as an Isolated Singularity

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