What is Complex analysis: Definition and 781 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. A

    Complex Contour Integral Problem, meaning

    Homework Statement First, let's take a look at the complex line integral. What is the geometry of the complex line integral? If we look at the real line integral GIF: [2]: http://en.wikipedia.org/wiki/File:Line_integral_of_scalar_field.gif The real line integral is a path, but then you...
  2. K

    Conformal mapping from polygon with circle segments

    I am looking for a conformal map from a "polygon" to eg the upper half plane, which consists of circle segments instead of lines. So for example, it could be a quadrilateral ABCD, but where AB is a circle segment. The closest I can find is the Schwarz-Christoffel mapping. Anyone has any tips?
  3. S

    Analytic verification of Kramers-Kronig Relations

    Homework Statement Show that the real and imaginary parts of the following susceptibility function satisfy the K-K relationships. Use the residue theorem. $$ \chi(\omega) = \frac{\omega_{p}^2}{(\omega_0^2-\omega^2)+i\gamma\omega} $$ Homework Equations The Kramers-Kronig relations are $$...
  4. F

    Functional Analysis vs. Complex Analysis?

    I have one slot to fill in in the coming term. The two candidates are Functional Analysis and Complex Analysis (both on the undergraduate level). Here are some questions: 1) Which one would you pick and why? 2) What other classes in the standard B.Sc. math curriculum rely on either of these...
  5. C

    Shoud I take Ring/Field Theory or Complex Analysis?

    Having just finished an introductory course on group theory (with some bits of ring and field theory), I am completely enthralled with this type of math. I initially planned on taking Complex Analysis next semester since so many people say it's "useful" for physics (this was also a compromise...
  6. A

    Why is the derivative of a polar function dy/dx?

    Homework Statement r = 2\cos(\theta) Homework EquationsThe Attempt at a Solution Hello, please do not evaluate. Why do textbook state that the derivative of the polar function (symbolic) is dy/dx and not dr/d\theta? It is a function of theta, then why is the derivative dy/dx? Idea: Even...
  7. V

    Allowed values for the "differentiability limit" in complex analysis

    In complex analysis differentiability for a function ##f## at a point ##z_0## in the interior of the domain of ##f## is defined as the existence of the limit $$ \lim_{h\rightarrow{}0}\frac{f(z_0+h)-f(z_0)}{h}.$$ But why are the possible ##z_0##'s in the closure of the domain of the original...
  8. A

    How to use contour, complex analysis to solve integrals?

    Homework Statement \int_{-\infty}^{\infty} \frac{\sin(x)}{x} using Complex Analysis Homework Equations Contour analysis on \int_{-\infty}^{\infty} \frac{\sin(x)}{x} The Attempt at a Solution Hello, I am completely new to contour integration. I would really appreciate it if someone can walk...
  9. W

    Complex analysis: residue integration question

    I'm asked to evaluate the following integral: \int_{c} \frac{30z^2-23z+5}{(2z-1)^2(3z-1)}dz where c is the unit circle. This function has a simple pole at z=\frac{1}{3} and a second order pole at z=\frac{1}{2}, both of which are within my region of integration. I then went about computing the...
  10. Darth Frodo

    Singularities Complex Analysis

    Homework Statement Determine the location and type of singularity of f(z) = 1/sin^2(z) Homework EquationsThe Attempt at a Solution I'm not really sure how to calculate this. At this point, we don't have explicit formulae for the coefficients of a Laurent series so I really don't know what to...
  11. M

    Are Complex Analysis and Complex Variables the same thing?

    Is Complex Analysis and Complex Variables the same thing? Is Complex Analysis pure or applied math? Is Complex Variables pure or applied math? What's the prerequisite of Complex Analysis and Complex Variables? Are they useful for the field of computer science?
  12. M

    More Complex Analysis studying recommendations

    Hi there, I'm currently taking Complex Analysis but do not feel like I have enough practice problems or course material (books, websites, Youtube channels, and etc) to study from. I was hoping some of you would have some stuff that I can check out. It would be greatly appreciated. I'm currently...
  13. M

    Complex Analysis: Learning Resources and Practice Problems

    Hi there, I was just wondering if anyone knows of any good materials, books, websites, Youtube users etc for me to teach myself Complex Analysis for school. Some good practice problems with answers and explanations would be wicked too. Thanks :)
  14. KleZMeR

    Evaluate Definite Integral with Complex Analysis

    Homework Statement I_1 = \int_0^{2\pi} \frac{sin\theta}{3+2cos\theta} d\theta Homework Equations Using identities to change from cos, sin, to variables of z, I get: 2iz^2 + 6iz + 2i in my denominator The Attempt at a Solution Looking for a singularity, will I use a quadratic...
  15. Darth Frodo

    Complex Analysis: Cauchy Riemann Equations 2

    Hi All, I was reading through Kreyzeig's Advanced Engineering Mathematics and came across two theorems in Complex Analysis. Theorem 1: Let f(z) = u(x,y) + iv(x,y) be defined and continuous in some neighborhood of a point z = x+iy and differentiable at z itself. Then, at that point, the...
  16. L

    Complex analysis - an integral with branch cuts

    Homework Statement Hi, I need to calculate the following integral: \int_{-\infty}^{+\infty}dx \frac{(\pi+\sqrt{x^2+m^2})^2(1+\cos x)}{(x^2-\pi^2)^2\sqrt{x^2+m^2}} The Attempt at a Solution I tried complexifying it: \oint dz...
  17. J

    Analyzing Analytic Functions: Solving a Complex Analysis Conundrum

    I came across an interesting problem that I have made no progress on. Let f be an analytic function on the disc ##D = \{z \in C ~|~ |z| < 1\}## satisfying ##f(0) = 1##. Is the following statement true or false? If ##f(a) = f^\prime(a) ## whenever ##\frac{1+a}{a}## and ##\frac{1-a}{a}## are...
  18. A

    Is complex analysis necessary for electrical engineering?

    I am and EE and CS double major and I am not sure whether to take complex analysis or not. Linear algebra, multivariable calculus, differential equations and probability are compulsory but complex analysis and stochastic processes are optional, so I am wondering whether I should take them or...
  19. S

    Why study different kinds of functions in Real and Complex Analysis?

    I've taken basic undergraduate Real and Complex Analysis, and I've noticed they focus on different kinds of functions. Real analysis studies things like Dirichlet and Cantor functions with infinitely many discontinuities while complex analysis studies mostly differentiable functions. My...
  20. Z

    MHB Tricky complex analysis questions....

    i. Let f and g be functions with a pole at c. Create rules (and prove them) about how we can combine f and g at c. and ii: Find the poles of the function : \frac{cotz+cosz}{sin2z} and classify these poles using part i.
  21. Z

    MHB Residues - complex analysis confusion.

    Hi! I am having a hard time with residues. :( I understand the formal definition of a residue, but anything past that and I am struggling. My course lecturer has a very confusing way of organising the course material and it is very hard to comprehend so was wondering if anyone could help with...
  22. D

    Complex Analysis: Solving z^2 = sqrt(z) & Mapping of Plane

    Homework Statement Let f(z) = sqrt(z) be the branch of the square root function with sqrt(z) = (r^1/2) (e^iΘ/2), 0≤Θ<2\pi, r > 0 (a) for what values of z is sqrt(z^2) = z? (b) Which part of the complex plane stretches, and which part shrinks under this transformation? Homework...
  23. 1

    Complex Analysis: Inverse Trig and Hyperbolic Functions Help

    Homework Statement I can't seem to get a few questions involving inverse trigonometric functions and hyperbolic functions. Here is one that I am stuck on: Evaluate the following in the form x+iy: sinh-1(i/2) = z Homework Equations sinh z = (ez - e-z)/2 The Attempt at a...
  24. D

    Complex Analysis: Solving Logarithmic Equations and Limits

    Hello, I'm solving the problems given in previous exams, and there's this question: Homework Statement a/ Give the value of ln(i), ln(-i) and i^i b/ If zo=-1-i , what is the value of lim [ ln(zo+e)-ln(zo+i*e) ] when e-> 0 Same question with zo=1+i Homework Equations The...
  25. N

    Complex analysis quick problem

    Homework Statement f(z) = u(x, y) + iv(x, y) where z ≡ x + iy. Let the fluid velocity be V = ∇u. If f(z) is analytic, show that df/dz = V_x − iV_y Homework Equations V_x = du/dx V_y = idu/dy The CR equations du/dx = dv/dy, du/dy = -dv/dx. The Attempt at a Solution I...
  26. D

    Complex analysis fourier series

    Hello, Homework Statement Develop in Fourier series 1/cos(z) and cotan(z) for Im(z)>0 Homework Equations The Attempt at a Solution I really don't know how to do this, i was looking at my notes and we just saw Fourier transform and there is no example for complex functions. I...
  27. S

    Real integrals using complex analysis

    Homework Statement After successfully solving a lot of integrals I gathered 4 ugly ones that I can not solve: a) ## \int _{-\infty} ^\infty \frac{cos(2x)}{x^4+1}dx## b) ##\int _0 ^\infty \frac{dx}{1+x^3}## c) ##\int _0 ^\infty \frac{x^2+1}{x^4+1}dx## d) ##\int _0 ^{2\pi } \frac{d\varphi...
  28. S

    Calculate real integrals using complex analysis

    Homework Statement Calculate real integrals using complex analysis a) ##\int_{-\infty}^{\infty}\frac{dx}{x^2+1}## b) ##\int_0^\infty \frac{sin(x)}{x}dx##Homework Equations The Attempt at a Solution a) ##\int_{-\infty }^{\infty }\frac{dz}{z^2+1}=\int_{-R}^{R}\frac{dx}{x^2+1}+\int...
  29. S

    MHB How Can Complex Analysis Be Used to Sum Powers of Sine Functions?

    How to find the sum using complex analysis sin^3x+sin^32x+sin^33x+sin^34x+...+sin^3nx
  30. T

    Real Analysis or Complex Analysis

    I'm about to start scheduling my courses for next year, and I have the option of taking either Real Analysis or Complex Analysis. I'm double majoring in Math and Physics, and I want to go to grad school to study either Applied Mathematics or Physics. I haven't taken any higher level math...
  31. N

    Finding Derivatives of Analytic Functions: Chain Rule Confusion

    Hello, I'm sorry if I'm not posting this to the correct place - this is my first post on PhysicsForums.com My question regards derivatives of analytic functions. Here it goes: Let w(z) = u(x,y) +iv(x,y) be an analytic function, where z = x + iy, for some x,y that are real...
  32. R

    (Complex analysis). Show that the inequality holds

    Homework Statement Show that the inequality\left|\frac{z^2-2z+4}{3x+10}\right|\leq3holds for all z\in\mathbb{C} such that |z|=2 Homework Equations Triangle inequality The Attempt at a Solution I'm not really sure how to go about this. the x is throwing me off. Should I write it out with...
  33. D

    Proving the Open Mapping Theorem for Continuous Functions on Complex Numbers

    Homework Statement Let a continuous function ##f:\mathbb{C}\rightarrow\mathbb{C}## satisfy ##|f(\mathbb{C})|\rightarrow\infty## as ##|z|\rightarrow\infty## and let ##f(\mathbb{C})## be an open set. Then ##f(\mathbb{C})=\mathbb{C}##. The Attempt at a Solution Suppose for contradiction that...
  34. phosgene

    Programs PDE's vs Complex analysis for physics/math major

    Hi guys. It's almost time to choose my courses for this year. I'm torn between taking PDE's due to how important it is for physics, or complex analysis due to just liking pure maths. If I do well enough, I'm *possibly* looking to do further study in mathematical physics. I was thinking that if...
  35. N

    Contour Integrals in complex analysis questions

    I am confused as to what we are obtaining when taking these contour integrals. I know that the close loop contour integral of a holomorphic function is 0. Is this analogous to the closed loop of integral of a conservative force which also gives 0? Also when I am integrating around a...
  36. L

    Complex Analysis - Taylor series of 1/(1+exp(z))

    Homework Statement Compute the first four terms of the Taylor series of \frac{1}{1+e^{z}} at z_{0} = 0 and give it's radius of convergence. Homework Equations e^{z} = \sum\frac{z^{n}}{n!} = 1 + z +\frac{z^{2}}{2!} + \frac{z^{3}}{3!} + o(z^{3}) \frac{1}{1+w} =...
  37. G

    Complex analysis question I don't understand solution

    So I am self studying complex analysis using some notes online so here he is trying to figure the map of certain domain applied to a function I follow everything but the end I don't get in the end when he got Re(w) < 1 how did he get that ? is it just because Re(w) which is 1/2 and 0 in this...
  38. N

    Easy complex analysis question

    [solved]Easy complex analysis question Hi. In the complex plain, since y = 0 (in z=x+iy) at the x axis, shouldn't the following be true? : ##y=0## \int_{-\infty}^{\infty} \frac{\cos(ax)}{x^2+2x+5} dx = \int_{-\infty}^{\infty} \frac{e^{iaz}}{z^2+2z+5} dz = \int_{-\infty}^{\infty} f(z) dz...
  39. S

    Quantity of zeros on a ring - Complex Analysis

    Determine the quantitiy of zeroes of the function: f(z)=z^{4}-8z+10 a) Inside the circle | z | < 1 b) Inside the ring 1 \leq | z | < 2a) f(z)=(z^{4}-8z)+10=g(z)+h(z) As |h(z)| \geq |g(z)| \forall z : | z | = 1 Then by Rouche's Theorem the number of zeros of the function inside the circle is the...
  40. G

    A gentle textbook of complex analysis?

    Is there a gentle textbook of complex analysis? Something equivalent to Larson's Calculus (or Stewart's). I have Schaum's Outline of Complex Variables (Spiegel-Lipschutz), and it's not bad.
  41. S

    Complex Analysis - Gaussian Function Integration

    I know that \displaystyle \int_{-\infty}^{\infty}e^{-x^{2}}dx=\sqrt{\pi} (You calculate the square of the integral, combine both integrals, change variable to polar coordinates and you can finally integrate that with ease). But in this exercise I have the following statament: Let be P_{R} the...
  42. N

    Complex Analysis pole-order problem - did I do it correctly?

    Homework Statement Determine the location of the singularities, including those at infinity. For poles also sate the order. f(z) = \frac{1}{(z+2i)^2}-\frac{z}{z-i}+\frac{z+i}{(z-i)^2} Homework Equations Theorem: If a function ##f(z)## has a zero of nth order at ##z_0##, then the function...
  43. N

    Complex Analysis: Is my proof of an easy theorem correct?

    Theorem: If a function f(z) has a zero of nth order at z0, then the function h(z)/f(z) has a pole of order n at z0 (where h(z) is analytic at ##z_0##). Can somebody explain this theorem for me? It isn't proved in my book because it's so "easy", but I don't get it? Is the sketch of the proof...
  44. F

    Differentiable functions in complex analysis

    Hello all, I have the following problem from Complex Analysis that I would like for someone to check my understanding on: Homework Statement The problem is to find the derivative if it exists of f(z) = \frac{e^{i\theta}}{r^2} = r^{-2}\cos \theta + i r^{-2}\sin \theta where I have already...
  45. F

    Solving integral in complex analysis

    Hi! I'm new here, been a fan of this site for years, but only now I felt the need of registering. Homework Statement Use the Residue Theorem to solve the integral: ∫[(cos(2t)) / (5-4*cos(t)) ] dt from t=0 to t=2pi2. The attempt at a solution I did a variable change z=eit. With that...
  46. B

    Complex Analysis and Mobius Transformation.

    Homework Statement If \phi \in \mathcal{M} (group of all linear fractional transformations or Mobius Transformations has three fixed points, then it must be the identity. (The proof should exploit the fact that \mathcal{M} is a group. The Attempt at a Solution Hi all, So...
  47. Y

    Complex analysis - partial fraction expansion

    Homework Statement Show that: Ʃ(-1)n/(n^2+a^2) (from n=0 to ∞) = pi/[asinh(pi*a)], a\neq in, n\in Z. Homework Equations f(z) = f(0) + Ʃbn(1/(z-an)+1/an) (from n=1 to ∞) , where bn is the residue of f(z) at an. The Attempt at a Solution The main problem is I don't how to pick the...
  48. D

    How can Laurent series be applied to complex analysis problems?

    Homework Statement Laurent series Homework Equations ##f(z) = sinh(z)## around origin The Attempt at a Solution ##sinh(z +\frac{1}{z}) = \sum_{-infty}^\infty A_nz^n## where ##A_n = \frac{1}{2\pi i} \oint \frac{sinh(z'+\frac{1}{z'})}{z'^{n+1}} d'## Let c = unit circle...
  49. G

    Harmonic Conjugates in Complex Analysis: Finding the Right Solution

    Find the harmonic conjugate of u. u = u(z) = ln(|z|) so u(z) = ln(sqrt(x^2 + y^2)) so basically I am trying to find now its harmonic conjugate I did all the math I got two solutions though one is v(z) = arctan(y/x) + C if I solve Au/Ax = -Au/Ay & other is v(z) = - arctan(x/y) + C if I...
  50. Q

    Complex analysis: I have to find sequence of C^inf functions that

    Homework Statement ... if fj are holomorphic on an open set U and fj \stackrel{uniformly}{\rightarrow} f on compact subsets of U then δ/δz(fj) \stackrel{uniformly}{\rightarrow} δ/δz(f) on compact subsets of U. Give an example to show that if the word "holomorphic" is replaced by "infinitely...
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