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. S

    Complex analysis continuity of functions

    Homework Statement The functions Re(z)/|z|, z/|z|, Re(z^2)/|z|^2, and zRe(z)/|z| are all defined for z!=0 (z is not equal to 0) Which of them can be defined at the point z=0 in such a way that the extended functions are continuous at z=0? It gives the answer to be: Only f(z)=zRe(z)/|z|...
  2. S

    Complex analysis limit points question

    Homework Statement Find the limit points of the set of all points z such that: a.) z=1+(-1)^{n}\frac{n}{n+1} (n=1, 2, ...) b.) z=\frac{1}{m}+\frac{i}{n} (m, n=+/-1, +/-2, ...) c.) z=\frac{p}{m}+i\frac{q}{n} (m, n, p, q=+/1, +/-2 ...) d.) |z|<1 Homework Equations None. The Attempt at a...
  3. A

    Integral Calculation with Complex Analysis - Can Residue Theorem Help?

    Hey guys. I need to calculate this integral so I was thinking about using the residue theorem. The thing is that the point 0 is not enclosed within the curve that I'm about to build, it's on it. Can I still use the theorem? Thanks a lot.
  4. S

    Prove: (z̄ )^k=(z̄ ^k) for z≠0 when k is negative

    Homework Statement Prove that (z̄ )^k =(z̄ ^k) for every integer k (provided z≠0 when k is negative) Homework Equations The Attempt at a Solution I let z=a+bi so, z̄ =a-bi Then I plugged that into one side of the equation to get (a-bi)^k I was going to try to manipulate this...
  5. B

    How to find the equation of a line in complex analysis?

    *This is not homework, though a class was the origin of my curiosity. In real analysis we could find the equation of a line that passes through two points by finding the slope and then plugging in one set of points to calculate the value of b. ie y = mx + b m = \frac{y_2-y_1}{x_2-x_1} In...
  6. S

    What Are the Loci of Points Satisfying Complex Inequalities in the Plane?

    Homework Statement #16)What are the loci of points z which satisfy the following relations...? d.) 0 < Re(iz) < 1 ? g.) α < arg(z) < β, γ < Re(z) < δ, where -π/2 < αα, β < π/2, γ > 0 ? I'm also wondering for help with this proof: #15)...Given: z_1 + z_2 + z_3 = 0 and |z_1| +...
  7. C

    Does anyone have a copy of Saff & Snider's Fundamentals of Complex Analysis ?

    Does anyone have a copy of Saff & Snider's "Fundamentals of Complex Analysis"? So I'm finally, as a graduate student, getting that last piece of undergraduate mathematics I missed: complex analysis. I enrolled in a class at the last minute, and wouldn't you know they assign homework for...
  8. C

    Fundamentals of Complex Analysis With Applications to Engineering and Science

    Text: Fundamentals of Complex Analysis With Applications to Engineering and Science by E.B. Saff and A.D. Snider I only ordered my textbook last week (yeah... I know), so I don't think it will get to me before my homework is due. Would some kind soul with this book please post these questions...
  9. A

    Complex Analysis Homework: Proving Equation w/ Cauchy's Integral Formula

    Homework Statement Hey guys. Look at this question please. I have two paths, and I need to proof the thing in red. They give a tip due, they say to show first that the equation in green is correct and then using Taylor development, to proof the red equation. I'm still in the first phase...
  10. D

    How Do You Find an Analytic Function Where the Argument is xy?

    1. This is something from complex analysis: Find the analytic function f(z)= f(x+iy) such that arg f(z)= xy. 2. w=f(z)=f(x+iy)=u(x,y)+iv(x,y) (*), w=\rho e^{i\theta} (**) Here are the Cauchy-Riemann conditions... \frac{\partial u}{\partial x}=\frac{\partial v}{\partial...
  11. I

    Ready for Complex Analysis Course: MAA 4402

    I was just wondering if I was ready to take a 4th year undergrad course in Complex Analysis. The book we will be using is called Complex Function Theory and its buy Sarason. I've taken a course in multivariable calculus, number theory, discrete mathematics, differential equations and modern...
  12. L

    Partial differentiation & complex analysis

    Homework Statement Let Δf= d^2f/dx^2+ d^g/dy^2 (laplace equation - Partial Derivatives) Show Δ(f(g(z))= Mod(g'(z))^2 * Δf(w,v) where g(z)=w(x,y)+v(x,y)i Homework Equations we propably need to use cauchy riemman equations: dw/dx = dv/dy and dw/dy = - dv/dx and chain rule The Attempt...
  13. R

    Real and Complex Analysis Textbook

    I'm currently looking for a textbook on Real and Complex Analysis. I currently own both Rudin's and Shilov's, and I'm interested to know if there are any more with that scope of topics. In English, please.
  14. C

    Complex Analysis - Contour Intergral

    Homework Statement The problem is to integrate: \oint_{C}\frac{dz}{z^{2}-1} C is a C.C.W circle |z| = 2. Homework Equations The Attempt at a Solution I used the Cauchy integral formula: \oint_{C}\frac{f(z)}{(z-z_{0})^{n+1}}dz = \frac{2 \pi i}{n!}f^{n}(z_{0}) Which...
  15. A

    Complex Analysis: Locus Sketching

    Homework Statement Sketch the locus of |z-2i|=z+3 in C 2. The attempt at a solution Let z=x+iy, then |z-i|=|x+iy-2i)|=|x+i(y-2)|=(x^2+(y-2)^2)^(1/2)=z+3 The problem is that I can't tell what this means geometrically. Is it a spiral?
  16. C

    Exponential Functions In Complex Analysis

    Can someone please tell me if I have the correct answer for this one? e^(5pi/4) = (1-i)/(-sqrt(2)) Thanks...
  17. C

    Finding the locus of points for complex analysis

    I think this should probably be easy, but I am stuck. My book is of no help. Find and describe the locus of points z satisfying the given equations: 1. |z-i|=Re z 2. |z-1|^2 =|z+1|^2 +6 I am thinking for the 1st one that I have to square both sides, but then what? What happens to...
  18. L

    Measures with Compact Support in Complex Analysis: Finiteness Assumptions

    I was reading in a book, says \mu is a measure with compact support K in C, meaning \mu(U)=0 for U\cap K=0.. Is \mu(K) assumed to be finite in this case? It doesn't say in the book, but they make a statement which is true if that's so. Is there usually some assumption about measures being...
  19. J

    Complex Analysis: Finding an Analytic Function for Re(z)=1-x-2xy

    hi I want to find an analytic funktion if Re(z) = 1 - x - 2xy My initial thought was to set U(x,y) = 1 - x - 2xy and then solve for V(x,y) through du/dx = dv/dy but it doesn't seem to go as far as I am concernd. Then I thought about the fact that Re(z) = (z + zbar)/2 and then work...
  20. T

    Complex analysis by Lars V. Ahlfors - how is that?

    Complex analysis : an introduction to the theory of analytic functions of one complex variable / [by] Lars V. Ahlfors. How do people find it?
  21. B

    Complex Analysis of a trigonometric function integral

    Homework Statement Find I = \int_0^{2\pi} \frac{1}{cos\phi+b} d\phi Homework Equations Given above.. The Attempt at a Solution This problem is an introductory problem to trigonometric functions and here is how the answer is obtained - but I have a question about it. First, here...
  22. V

    A graduate level question in complex analysis

    If f and g are two entire functions such that mod(f(z)) <= mod(g(z)) for all z in C, prove that f=cg for some complex constant c.
  23. M

    Complex analysis of electrostatic problem

    I'm not sure whether to post this in the Mathematics or Physics forums, but I figure this problem is easily reduced to its transformation irrespective of the physics it describes. Consider a semi-infinite sheet of (infinitely thin) conductor charged to a potential V. It is placed at a distance...
  24. B

    Book recs please - complex analysis, riemann surfaces, multi-valued functions

    Hi everyone, hope this is the right place to put this :) I have just finished "Theory of Functions" Vol. 1 & 2 by Konrad Knopp. I'd like to continue with a book that picks up where the second volume it left off. (Especially would be nice is a more "modern" book) The second volume is about...
  25. L

    Recommended Complex Analysis Books for Self-Study: A Scientist's Perspective

    I've never had any complex analysis, but I'd like to teach myself. I don't know of any good books though. I learned Real Analysis with Pugh, so I'd like a Complex Analysis book on a similar level (or maybe higher). I.e., I'm looking for a book that develops Complex Numbers and functions...
  26. A

    Question on linear combinations of sines and cosine (complex analysis)

    I have a question on complex analysis. Given a differential equation, \dfrac{d^2 \psi}{dx^2} + k ^2 \psi = 0 we know that the general solution (before imposing any boundary conditions) is, \psi (x) = A cos(kx)+B sin(kx). Now here's something I don't quite understand. The solution...
  27. J

    Complex Analysis material question

    I'm an undergraduate studying mathematics. I did really well in differential equations and abstract algebra, but struggled with our course "Analysis I." I'm taking complex analysis next spring (here's a description of the course, but I'm sure it's not much different than any other complex...
  28. T

    Complex Analysis: Proving Bounds for |e^z-1|

    Hi. I need to show that for all |z|\leq1 : (3-e)|z|\leq|e^{z}-1|\leq(e-1)|z| Now...
  29. N

    Complex analysis - Cauchy estimates

    Homework Statement Let D\subset\mathbb{C} be the unitdisc and F=\{f:D\rightarrow D\,|\,\forall z\in D\partial_{\bar{z}}f=0\}, calculate L=\sup_{f\in F}|f''(0)|. Show that there is an g\in F with g''(0)=L. I am a bit stuck. But I think that it might be an idea to start with Cauchy estimate. Any...
  30. M

    Complex Anal. Problems: Need Help!

    I have the following problems (1)Consider the series ∑z^n,|z|<1 z is in C I thik this series is absolutely and uniformly comvergent since the series ∑|z|^n is con vergent for |z|<1,but I have a book saying that it is absolutely convergent,not uniformly...i am confused... (2)for the function...
  31. R

    Complex analysis - maximum modulus &amp; analytic function

    [SOLVED] complex analysis - maximum modulus &amp; analytic function Hi all, I'm having difficulty figuring out how to do the following two problems in complex analysis. I need help! 1. Consider the infinite strip -\pi< I am z < \pi. Does maximum modulus principle apply to this strip? Why or...
  32. J

    Complex Analysis: Solve Injective Function f(z)=az+b

    [SOLVED] Complex Analysis PROBLEM Let a function f be entire and injective. Show that f(z)=az+b for some complex numbers a,b where a is not 0. Hint: Apply Casorati-Weierstrass Theorem to f(1/z). THEOREM Casorati-Weierstrass Theorem: Let f be holomorphic on a disk D=D_r(z_0)\{z_0} and have an...
  33. S

    Integral of (1/8z^3 -1) around Contour C=1: Step by Step Guide

    Also when trying to find the integral of (1/8z^3 -1) around the contour c=1. I found the singularities to be 1/2, 1/2exp(2pi/3), and 1/2exp(4pi/3) What is the next step here. Do I just assume the integral is 6pi(i) after using partial fractions to find the numerators of the 3 fractions...
  34. M

    Fourier series via complex analysis

    Homework Statement Show that f is 2-pi periodic and analytic on the strip \vert Im(z) \vert < \eta, iff it has a Fourier expansion f(z) = \sum_{n = -\infty}^{\infty} a_{n}z^{n}, and that a_n = \frac{1}{2 \pi i} \int_{0}^{2\pi} e^{-inx}f(x) dx. Also, there's something about the lim sup of...
  35. S

    Complex analysis zero/branch problem

    Homework Statement Let f be a holomorphic function in the open subset G or C. Let the point Z of G be a zero of f of order m. Prove that there is a branch of f^(1/m) in some open disk centered at Z Homework Equations Branch- a continuous function g in G such that, for each x in G, the...
  36. S

    The identity theroem complex analysis

    Homework Statement Prove that there is no holomorphic function f in the open unit disk such that f(1/n)=((-1)^n)/(n^2) for n=2,3,4... Homework Equations The identity theorem: Let f and g be holomorphic functions in the connected open subset of C, G. If f(z)=g(z) for all z in a subset...
  37. U

    Analytic Functions and Complex Analysis: Understanding the Relationship

    I don't really know which forum to post this in but I just have a quick question: Is it sufficient to say that a function is analytic on a domain if it has a derivative and the derivative is continuous?
  38. MathematicalPhysicist

    Complex analysis question (Roche theorem)

    Let R be domain which contains the closed circle: |z|<=1, Let f be analytic function s.t f(0)=1, |f(z)|>3/2 in |z|=1, show that in |z|<1 f has at least 1 root, and and one fixed point, i.e s.t that f(z0)=z0. now here what I did, let's define g(z)=f(z)-z, and we first need to show that the...
  39. M

    Connectivity of Complex Analysis Polynomial Sets | Degree n+1

    Homework Statement Let p(z) be a polynomial of degree n \geq 1. Show that \left\{z \in \mathbb{C} : \left|p(z)\right| > 1 \right\}[/tex] is connected with connectivity at most n+1. Homework Equations A region (connected, open set) considered as a set in the complex plane has finite...
  40. J

    Complex Analysis: Show RHS=LHS for Real r?

    [SOLVED] Complex Analysis Show that \mbox{Re}\left(\frac{Re^{i\theta}+r}{Re^{i\theta}-r}\right)=\frac{R^2-r^2}{R^2-2Rr\cos\theta+r^2} where R is the radius of a disc. I was able to show this for all real values of r. However, the problem doesn't specify whether r is real or complex. After...
  41. quantumdude

    Multivariable Complex Analysis: Uses in physics?

    I think this is the first time I've used this forum for myself. :approve: OK, I'm picking out courses for next semester. Right now I'm in the second semester of Complex Analysis (based on Serge Lang's book) which is a grad level course in single variable complex analysis. My school offers a...
  42. I

    Quick complex analysis (integration) question

    I want to show that the integral from -1 to 1 of z^i = (1-i)(1+exp(-pi)/2 where the path of integration is any contour from z=-1 to z=1 that lies above the real axis (except for its endpoints). So, I know that z^i=exp(i log(z)) and the problem states that |z|>0, and arg(z) is between -pi/2...
  43. M

    Probably obvious complex analysis question

    Homework Statement \int_{|z| = 2} \sqrt{z^2 - 1} Homework Equations \sqrt{z^2 - 1} = e^{\frac{1}{2} log(z+1) + \frac{1}{2} log(z - 1)} The Attempt at a Solution Honestly, my only thoughts are expanding this as some hideous Taylor series and integrating term by term. But I know...
  44. F

    Complex analysis: having partials is the same as being well defined?

    Complex analysis: having partials is the same as being "well defined?" My professor proved this theorem in class and I don't know if I even wrote it down correctly in my notes. I don't have access to the book so I need to know if this makes sense. Here is the theorem: Under these conditions...
  45. M

    Complex Analysis Fun: Analytic Antiderivatives in {z:|z|>2}

    Homework Statement Show that \frac{z}{(z-1)(z-2)(z+1)} has an analytic antiderivative in \{z \in \bold{C}:|z|>2\}. Does the same function with z^2 replacing z (EDIT: I mean replacing the z in the numerator, not everywhere) have an analytic antiderivative in that region? Homework...
  46. E

    Is the Complex Analysis Problem with \(\sqrt{z}\) on the Unit Circle Ambiguous?

    Homework Statement Evaluate \int_{\gamma} \sqrt {z} dz where \gamma is the upper half of the unit circle. I contend that this problem does not make sense i.e it is ambiguous because they did not tell us specifically what branch of the complex square root function to use. Am I right?Homework...
  47. E

    Please recommend a complex analysis book for The road to reality

    Please recommend a complex analysis book for "The road to reality" Guys I am a electrical engineer who studied calculus III about 15 years ago. That time I memorized formulas to pass exams and never have much of a understanding of complex analysis. Never touched high math again after...
  48. L

    Programs Complex analysis as physics major

    I am a physics major and I have taken many math courses, but not Complex Variables. I did a little contour integration along time ago, but I never took it as a course. I do, however, have the option to take this semester. Should I take it instead of another physics elective? I know that it is...
  49. Shaun Culver

    Advice on complex analysis, Riemann surface & complex mappings.

    Could anybody please give advice for the study of complex analysis, Riemann surfaces & complex mappings. These subjects form the content of chapters 7 & 8 of Roger Penrose's "The Road to Reality". Any advice will do: maybe suggestions on books to supplement the learning, or books to further my...
  50. I

    Solving Complex Analysis: Finding Points |z-1|=|z+i|

    I am to find all plints z in the complext plane that satisfies |z-1|=|z+i| The work follows: let z=a+bi |a+bi-1|=|a+bi+i| (a-1)^2+b^2=a^2+(b+1)^2 a^2-2a+1+b^2=a^2+b^2+2b+1 -a=b the correct answer should be a perpendicular bisector of segments joining z=1 and z=-i my result looks...
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