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. MakVish

    Complex Analysis: Find Analytic Functions w/ |ƒ(z)-1| + |ƒ(z)+1| = 4

    Homework Statement Find all analytic functions ƒ: ℂ→ℂ such that |ƒ(z)-1| + |ƒ(z)+1| = 4 for all z∈ℂ and ƒ(0) = √3 i The Attempt at a Solution I see that the sum of the distance is constant hence it should represent an ellipse. However, I am not able to find the exact form for ƒ(z). Any help...
  2. Santilopez10

    How Does the Sinc Function Integral Relate to Quantum Collision Theory?

    Homework Statement The following is a problem from "Applied Complex Variables for Scientists and Engineers" It states: The following integral occurs in the quantum theory of collisions: $$I=\int_{-\infty}^{\infty} \frac {sin(t)} {t}e^{ipt} \, dt$$ where p is real. Show that $$I=\begin{cases}0 &...
  3. O

    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...
  4. S

    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)...
  5. e0ne199

    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...
  6. A

    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...
  7. Baibhab Bose

    A Understanding the Order of Poles in Complex Functions

    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...
  8. A

    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...
  9. T

    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...
  10. T

    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...
  11. G

    Complex analysis question

    Hey, I have been stuck on this question for a while: I have tried to follow the hint, but I am not sure where to go next to get the result. Have I started correctly? I am not sure how to show that the integral is zero. If I can show it is less than zero, I also don't see how that shows it...
  12. T

    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...
  13. T

    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...
  14. T

    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##...
  15. Measle

    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...
  16. Measle

    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
  17. Measle

    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...
  18. Safder Aree

    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...
  19. D

    I Complex analysis - removable singular points

    Hi. I have 2 questions regarding removable singular points. 1 - the residue at a removable singularity is always zero so by the residue theorem the integral around a closed simple contour is zero. Cauchy's theorem states the integral around a simple closed contour for an analytic function is...
  20. Q

    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...
  21. T

    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...
  22. T

    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...
  23. A

    I Verifying Equality: \mathcal{Im}[A+B+Te^{2ip}]=0

    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...
  24. 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

    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
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  25. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Recalling Singularities of Analytic Functions: Non-isolated and Isolated Removable

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Recalling Singularities of Analytic Functions: Non-isolated and Isolated Removable

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

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

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

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

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

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

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  29. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Studying Infinity: Formulating Epsilon-Delta Definitions for Infinite Limits

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Studying Infinity: Formulating Epsilon-Delta Definitions for Infinite Limits

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  30. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- When is a function analytic at infinity ?

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- When is a function analytic at infinity ?

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  31. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Laurent Expansion at Infinity and Riemann's Removable Singularities Theorem

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Laurent Expansion at Infinity and Riemann's Removable Singularities Theorem

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  32. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- The Generalized Liouville Theorem: Little Brother of Little Picard

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- The Generalized Liouville Theorem: Little Brother of Little Picard

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  33. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Morera's Theorem at Infinity, Infinity as a Pole and Behaviour at Infinity

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Morera's Theorem at Infinity, Infinity as a Pole and Behaviour at Infinity

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  34. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Residue at Infinity and Introduction to the Residue Theorem for the Extended

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Residue at Infinity and Introduction to the Residue Theorem for the Extended

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  35. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Proofs of Two Avatars of the Residue Theorem for the Extended Complex Plane

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Proofs of Two Avatars of the Residue Theorem for the Extended Complex Plane

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  36. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Infinity as an Essential Singularity and Transcendental Entire Functions

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Infinity as an Essential Singularity and Transcendental Entire Functions

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  37. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Meromorphic Functions on the Extended Complex Plane

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Meromorphic Functions on the Extended Complex Plane

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  38. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- The Ubiquity of Meromorphic Functions

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- The Ubiquity of Meromorphic Functions

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  39. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Continuity of Meromorphic Functions at Poles and Topologi

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Continuity of Meromorphic Functions at Poles and Topologi

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  40. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Why Normal Convergence, but Not Globally Uniform Convergence

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Why Normal Convergence, but Not Globally Uniform Convergence

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  41. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Measuring Distances to Infinity, the Function Infinity and Normal Convergence

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Measuring Distances to Infinity, the Function Infinity and Normal Convergence

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  42. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- The Invariance Under Inversion of the Spherical Metric on the Extended Complex Plane

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- The Invariance Under Inversion of the Spherical Metric on the Extended Complex Plane

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  43. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Introduction to Hurwitz's Theorem for Normal Convergence of Holomorphic Functions

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Introduction to Hurwitz's Theorem for Normal Convergence of Holomorphic Functions

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  44. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Completion of Proof of Hurwitz's Theorem for Normal Limits of Analytic Functions

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Completion of Proof of Hurwitz's Theorem for Normal Limits of Analytic Functions

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  45. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Hurwitz's Theorem for Normal Limits of Meromorphic Functions in the Spherical Metric

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Hurwitz's Theorem for Normal Limits of Meromorphic Functions in the Spherical Metric

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  46. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- What could the Derivative of a Meromorphic Function

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- What could the Derivative of a Meromorphic Function

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  47. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Defining the Spherical Derivative of a Meromorphic Function

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Defining the Spherical Derivative of a Meromorphic Function

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  48. Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Well-definedness of the Spherical Derivative of a Meromorphic Function

    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Well-definedness of the Spherical Derivative of a Meromorphic Function

    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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    Advanced Complex Analysis - Part 2 by Dr. T.E.V. Balaji (NPTEL):- Topological Preliminaries: Translating Compactness into Boundedness

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

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    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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