Complex analysis Definition and 784 Threads

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. 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...
  2. 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...
  3. 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...
  4. G

    Mapping of a Complex Region Using a Rational Function

    Homework Statement Let S = {z : 1<= Im(z) <=2}. Determine f(S) if f: S ->C defined by f(z) = (z + 1) / (z - 1)Homework Equations z = x + iy The Attempt at a Solution [attempt at solution] so here my solution f(z) = 1 + 2/(z - 1) after doing some algebra <-> f(z) = x^2 + y^2/((x - 1)^2 +...
  5. H

    Contour Deformation and Jordan's Lemma in Complex Analysis

    Lets say you're doing one of those integrals from -\infty to \infty on the real axis and you chose to do it by contour integration. Let's say your integral is one of those integrals that's resolved by using Jordan's lemma. If you close the contour by making a giant loop such that Jordan's lemma...
  6. djh101

    Mathematical Methods or Complex Analysis?

    Hi, Physics Forums. I just have a quick question regarding which two math-type electives I should take as a physical chemistry major. Right now I am enrolled in Linear Algebra (Math 115A) and Mathematical Methods for Physicists (Physics 131) and plan on taking the second MM (Physics 132) next...
  7. U

    Complex analysis, construct analytic f given Re(f)

    Homework Statement u(x,y) = sin(x^2-y^2)cosh(2xy) Find a function f(x+iy) = u(x,y) + iv(x,y), where v(x,y) is a real function, such that f is analytical in all of the complex plane. Find all such f. The attempt at a solution I expanded using Euler's for sin and cosh which gave me u(x,y) =...
  8. N

    Complex Analysis: Find f'(z) & Region of Analyticity

    Homework Statement For each of the following functions f(z), find f'(z) and identify the maximal region for which f(z) is analytic. 1. f(z)=1/(z^2+1) 2. f(z)=e^{-1/z} Homework Equations The Attempt at a Solution 1. f'(z)=\frac{-2z}{(z^2+1)^2} <--this part is easy. I'm having...
  9. B

    Complex Analysis, Complex Differentiable Question

    Homework Statement Define f : \mathbb{C} \rightarrow \mathbb{C} by f(z) = \left \{ \begin{array}{11} |z|^2 \sin (\frac{1}{|z|}), \mbox{when $z \ne 0$}, \\ 0, \mbox{when z = 0} . \end{array} \right. Show that f is complex-differentiable at the origin although the...
  10. L

    This looks almost too easy where did I go wrong? Complex Analysis.

    Homework Statement http://www.math.northwestern.edu/graduate/prelims/AnalysisPrelim2010FallFinalVersion.pdf Problem 2 of Part III. Homework Equations Complex Analysis. The Attempt at a Solution So, I think my proof is wrong (since I never used the fact that it was f^2) as opposed to f...
  11. C

    Real Analysis and Complex Analysis

    I was wondering if it is too ambitious to take both Real Analysis and Complex Analysis in the same semester. Thanks.
  12. S

    Complex analysis book with lots of solved problems?

    Hi. I am taking complex analysis over the summer and I am having a difficult time learning the concepts. I've tried reading several dfiferent textbooks, and though they sometimtes state the same theorem using different wording, different arguments, etc, I am still having a hard time...
  13. caffeinemachine

    MHB Which Book Should I Choose for Starting Complex Analysis?

    Hello MHB, I want to start reading Complex Analysis. I have never read any textbook on this subject. I have good background in Algebra, Linear Algebra, Point-set Topology and Real Analysis. Right now I want to prepare for the subject GRE in Mathematics so please suggest a book keeping that i...
  14. S

    Should I have learned real analysis before taking complex analysis

    Hi I am taking summer class in complex analysis and I am having a horrible time. I don't understand anything we've covered so far, e.g. Cauchy-Goursat theorem, Laurent series, series expansion, etc. The prereqs was just Calc III, which I got an A- in. The textbook isn't much help...
  15. Mandelbroth

    Proving a conjecture in complex analysis

    How would I go about proving that, for a curve in the complex plane ##\alpha## and a real number ##\beta##, $$\exists\alpha,\beta: \frac{x}{2\pi i}\int\limits_\alpha \frac{\Gamma(z+\frac{1}{2})\Gamma(-z)x^{\beta z}}{\Gamma(\frac{3}{2}-z)}\, dz = \arctan{x}?$$ The poles of the integrand are...
  16. I

    Complex Analysis - Rational Functions

    Homework Statement I'm studying for my final exam and came across this problem: Let f and g be entire analytic functions and |f(z)|<|g(z)| when |z|>1. Show that f/g is a rational function. The Attempt at a Solution I really have no clue where to go :(
  17. N

    Complex analysis proof with residue theorem, argument principle

    Homework Statement Let C be a regular curve enclosing the distinct points w1,..., wn and let p(w)= (w-w1)(w-w2)...(w-wn). Suppose that f(w) is analytic in a region that includes C. Show that P(z)= (1/2\pii)∫(f(w)\divp(w))\times((p(w)-p(z)\div(w-z))\timesdw is a polynomial of degree n-1...
  18. Hercuflea

    Are black holes explained by complex analysis?

    Is the center of a black hole essentially a pole, or a "point at infinity"? I always thought about this in my complex analysis class because one variable complex functions are 4 dimensional, which could translate into space-time. Black holes have to have infinite density in their center, too...
  19. Fernando Revilla

    MHB Billy 's question at Yahoo Answers (Complex analysis)

    Here is the question: Here is a link to the question: Complex analysis help? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  20. T

    MHB A second complex analysis question

    So this time I have to solve cos(z)=2i My approach: cos(z)= [ e^(iz) + e^(-iz) ] / 2 = 2i Rearranging and setting e^(iz) = w we get a quadratic w^2 - 4iw + 1 = 0 The quadratic yields two solutions: w=e^(iz) = i(2 + sqrt(5)) or e^(iz) = i(2-sqrt(5))And now my problem is here. In the...
  21. T

    MHB Solving Complex Analysis: Find z for z^(1+4i) = i

    Hello guys! I have to find for z in D= C \ {x in R: x<=0} with z^(1+4i) = i a) all possible values of Log(z) b)all possible values of z.Now my approach is: Write z^(1+4i) = exp((1+4i) * Logz) = i = exp(i*pi/2) which holds iff (1+4i) * Logz = i*pi/2 + i*k*2*pi where k in Z.After a lot of...
  22. N

    Integral question for cos(2z) in complex analysis

    Homework Statement Evaluate ∫cos(2z)dz from pi/2 to pi/2+i Homework Equations The Attempt at a Solution I know the cos function is entire, thus independent of path and I just need to evaluate the end points. I also know when you integrate you get (1/2)sin2z. My only question is...
  23. N

    Complex analysis help please (just a quadratic proof)

    Homework Statement Let A be complex, B be real. Show \left|z^2\right|+ Re(Az)+B=0 only has a solution if and only if \left|A^2\right|\geq4B. Then, assuming the above condition holds, show the solution is a circle or a single point. Homework Equations General quadratic equation I think...
  24. G

    Proving mapping must be a rational function (complex analysis)

    Homework Statement Let f(x) be a function which is defined in the open unit disk (|z| < 1) and is analytic there. f(z) maps the unit disk onto itself k times, meaning |f(z)| < 1 for all |z| < 1 and every point in the unit disk has k preimages under f(z). Prove that f(z) must be a rational...
  25. K

    Complex Analysis Practice Problems

    Homework Statement a) Find the radius of convergence of the following complex series and the complex point, where the center of the disk of convergence is located: \sum_{n=1}^{inf} 4^n (z-i-5)^{2n} b) Find the Laurent series of the following function, f(z), about the singularity, z = 2, and...
  26. I

    Does Convergence of Sum(z_j) and Sum((z_j)^2) Imply Convergence of Sum(|z_j|^2)?

    Homework Statement Assume that z_j is a sequence where j indexes from 1 to infinity are in the complex numbers such that the real part of z > 0. Is it true or false that if sum(z_j) and sum_((z_j)^2) both converge then sum(|z^j|^2) also converges? Homework Equations The Attempt at...
  27. K

    Is There an Easier Way to Prove S is Disconnected?

    Homework Statement Let S={zεℂ: |z|<1 or |z-2|<1}. show that S is not connected.Homework Equations My prof use this definition of disconnected set. Disconnected set - A set S \subseteqℂ is disconnected if S is a union of two disjoint sets A and A' s.t. there exists open sets B and B' with A...
  28. N

    Complex analysis topic help Need explanation

    Homework Statement This isn't a specific problem, but more of a type of problem I do not get. I am taking undergrad complex analysis, using the book by Bak and Newman. Its only a couple week in and I am having to spend a lot of time on it, last week I spent about 7 hours on the homework (which...
  29. M

    Complex Analysis: Find 2 Square Roots, Solve Eqn, Form Triangle

    Homework Statement 1- Find the two square roots of the complex number z=3+4i. 2a- Solve in ℂ the equations: (E): 4z^2-10iz-7-i=0 b- Let a and b be solutions to (E) such that: Re(a)<0 and the two points A and B plots/pictures of a and b. Show that b/a=1-i. Conclude that AOB is an...
  30. micromass

    Analysis A Collection of Problems on Complex Analysis by Volkovyskii, Lunts, Aramanovich

    Author: L. I. Volkovyskii, G. L. Lunts, I. G. Aramanovich Title: A Collection of Problems on Complex Analysis Amazon link: https://www.amazon.com/dp/0486669130/?tag=pfamazon01-20 Table of Contents: Foreword Complex numbers and functions of a complex variable Complex numbers (complex...
  31. Z

    Schools Biophysics/Physics Graduate School w/o Complex Analysis

    Hey y'all, I'm in my last semester about to obtain my physics degree, and I am very interested in biophysical research or energy research. With that in mind I have my eyes set on biophysics programs and straight physics programs. The concern I have is that for my last semester I can only take...
  32. micromass

    Analysis Complex Analysis by Stein and Shakarchi

    Author: Elias Stein, Rami Shakarchi Title: Complex Analysis Amazon Link: https://www.amazon.com/dp/0691113858/?tag=pfamazon01-20 Prerequisities: Fourier Analysis by Stein, Shakarchi Level: Undergrad Table of Contents: Foreword Introduction Preliminaries to Complex Analysis Complex...
  33. micromass

    Analysis Master Complex Analysis with Serge Lang: Prerequisites & Techniques for Grads

    Author: Serge Lang Title: Complex Analysis Amazon Link: https://www.amazon.com/dp/0387985921/?tag=pfamazon01-20 Prerequisities: Basic analysis Level: Grad Table of Contents: Foreword Prerequisites Basic Theory Complex Numbers and Functions Definition Polar Form Complex Valued...
  34. micromass

    Analysis Real and Complex Analysis by Rudin

    Author: Walter Rudin Title: Real and Complex Analysis Amazon Link: https://www.amazon.com/dp/0070542341/?tag=pfamazon01-20 Prerequisities: Baby Rudin Level: Grad Table of Contents: Preface Prologue: The Exponential Function Abstract Integration Set-theoretic notatons and terminology...
  35. alyafey22

    MHB Solve the following integral without complex analysis:

    \[\int^{\infty}_{0} \frac{\cos(x) }{1+x^2}\,dx\] I know it can be solved by Fourier transform and also by residues , but my teacher asked me to solve it by not using transformation or complex analysis (Happy)
  36. Greg Bernhardt

    Analysis Basic Complex Analysis by by J. E. Marsden and M.J. Hoffman

    Author: by J. E. Marsden and M.J. Hoffman Title: Basic Complex Analysis Amazon Link: https://www.amazon.com/dp/071672877X/?tag=pfamazon01-20 Prerequisities: Table of Contents: Analytic Functions Introduction to Complex Numbers Properties of Complex Numbers Some Elementary Functions...
  37. Greg Bernhardt

    Analysis Visual Complex Analysis by Tristan Needham

    Author: Tristan Needham Title: Visual Complex Analysis Amazon Link: https://www.amazon.com/dp/0198534469/?tag=pfamazon01-20 Prerequisities: Contents: Table of Contents: Geometry and Complex Arithmetic Introduction Historical Sketch Bombelli's "Wild Thought" Some Terminology and...
  38. P

    Complex Analysis - Fibonacci Identity

    Hey guys~ I was looking for a way to derive a formula for fn (the nth term in the fibonacci sequence). While looking for this, I came across a potential solution using the residue theorem. Using the generating function Ʃk≥0 fnzn, find the identity for fn. The problem looks like the right...
  39. R

    Complex Analysis - Cauchy Integral? Which technique do I use?

    Homework Statement \int_0^\infty\frac{x^{p-1}}{1+ x}dx ** I could not get p-1 to show as the exponent; the problem is x raised to the power of p-1. \int_0^\infty\frac{ln(x) dx}{(x^2+1)^2} The Attempt at a Solution There is no attempt, but I would like to make one! I'm asking...
  40. T

    MHB Complex Analysis Review Question

    Use residues toe evaluate the improper integral Use residues toe evaluate the improper integral $$\int_{0}^{\infty} \dfrac{dx}{(x^2 +9)^3}.$$ Explain all steps including convergence. No need to simplify the final answer. I took this off a old mid-term that I was looking at, no solution is...
  41. M

    Complex analysis integration. Strange result

    Hi, Consider the real variable x and some real constant x_{0}. I want to integrate \int_{-\infty}^{\infty}\frac{x_{0}}{x_{0}-x} This blows up when the denominator is zero but we can still take the principal value of the integral. That is, we notice that the integral is an odd function around...
  42. J

    Complex analysis- show that the integral of this function exists

    Hi, I just had my exam on complex analysis and would just like to know if I did this question correctly. It said that the function f(z) was analytic and to show that the integral of f(z)-\frac{c}{z} existed for some constant c, then to find a formula for c in term of an integral of f(z). I...
  43. H

    Schools Can High School Students Study Complex Analysis Independently?

    I am currently a freshman in high school in the United States. I am very interested in mathematical and theoretical physics. For about the past year or two I have been studying on my own. Over the last summer I spent most of July and August reading Calculus by James Stewart and feel that I...
  44. S

    Introductory Complex Analysis - Cartan?

    I would like a thorough but not overly comprehensive intro text for complex analysis. My background is one variable real analysis (Rudin), Linear Algebra (Friedberg), Abstract Algebra (Herstein). I know only basic point-set topology (from Rudin), and I haven't dealt at all with differential...
  45. Avatrin

    Complex Analysis and vector calculus

    Hi How much different is complex analysis from vector calculus? To me complex analysis looks like vector calculus combined with algebra of complex numbers..
  46. J

    Complex analysis - Louvilles Theorem (I think)

    Hi I am doing a past exam for my complex analysis course and I should just mention right now that while it's a mix of pure/applied math I have never done a pure math unit before and i really really really suck at doing proofs and such.. Given c>0 and f(z) is entire such that |f(z)| ≤ c|z|...
  47. F

    Complex Analysis Solutions: Real and Imaginary Parts

    Find the real part and imaginary part of the following exercises. 1) w = ((e^(conjugated(z)))^2 2) w = tgz Solutions: 1) u= (e^(x^2-y^2))cos2xy v= -(e^(x^2-y^2))sin2xy 2) u= (sinxcosx)/(ch^2y-sin^2x) v= (shychy)/(ch^2-sin^2x) -------------------------------------...
  48. B

    Mapping of Functions (Complex Analysis)

    Homework Statement Show that the function w = e^z maps the shaded rectangle in Fig X one-to-one onto the semi-annulus in Fig y. Fig x is the rectangle -1<x<1 ; 0<y<(x+pi(i)) Fig y is the semi-annulus such that y>0 and -e<r<-1/e Homework Equations ... The Attempt at a...
  49. C

    Branch cuts in complex analysis

    Homework Statement Given that the standard square root sqrt(anything) has a branch cut from (-inf,0), find the branch cuts of the following: z+sqrt(z^2-1) z+isqrt(1-z^2) z+sqrt(z+1)sqrt(z-1)Homework Equations The Attempt at a Solution I understand what branch cuts do (multivalue functions ->...
  50. R

    Proving the Circle Property of Infinite Sequence in Complex Analysis

    Homework Statement To prove that the sequence a_{n}= \prod_{k}^\infty (1 + \frac{i}{k}) when n is infinite constitutes points on a circle.Homework Equations Ehh no idea what equations shall be used.The Attempt at a Solution A friend asked me this, but I am usually engaged more with the physical...
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