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. P

    Complex Analysis: Sums of elementary fractions

    I have a homework question that reads: Represent the following rational functions as sums of elementary fractions and find the primitive functions ( indefinite integrals ); (a) f(z)=z-2/z^2+1 But my confusion arrises when I read sums of elementary fractions. I think what the question is...
  2. D

    Is this Complex Function Continuous?

    Homework Statement f:Complex Plane ->Complex Plane by f(z) = (e^z - z^e)/(z^3-1) continuous? (Hint: it has more than one discontinuity.) The Attempt at a Solution My attempt at a solution was thus, initially I expanded z^3 and tried to find where it equaled 1. That wasn't...
  3. P

    Complex Analysis: Holomorphic functions

    So my teacher explained what holomorphic functions were today. But it did not make much sense. As I am attempting to do my homework, I realized that I still don't really know what a Holomorphic function is, let alone how to show that one is. The questions looks like this: show that...
  4. S

    Isolated singularity (complex analysis)

    Homework Statement 1) \frac{e^{z}-1}{z} Locate the isolated singularity of the function and tell what kind of singularity it is. 2) \frac{1}{1 - cos(z)} z_0 = 0 find the laurant series for the given function about the indicated point. Also, give the residue of the function at the...
  5. M

    Complex Analysis Integral

    Homework Statement Evaluate \oint_C \f(z) \, dz where C is the unit circle at the origin, and f(z) is given by the following: A. e^{z}^{2} (the z2 is suppose to be z squared) B. 1/(z^{2}-4) Homework Equations The Attempt at a Solution I'm completely confused
  6. S

    Determing where function is differentiable (Complex Analysis)

    Homework Statement Determine where the function f has a derivative, as a function of a complex variable: f(x +iy) = 1/(x+i3y) The Attempt at a Solution I know the cauchy-riemann is not satisfied, so does that simply mean the function is not differentiable anywhere?
  7. S

    Two complex analysis questions

    Homework Statement 1) Where is f(z)=\frac{sin(z)}{z^{3}+1} differentiable? Analytic? 2) Solve the equation Log(z)=i\frac{3\pi}{2} Homework Equations none really... The Attempt at a Solution For #1 I started out trying to expand this with z=x+iy, but it got extremely messy...
  8. H

    Is Real Analysis Necessary Before Complex Analysis?

    Just wondering, when starting on introductory analysis is it logical to do real analysis before complex variables? My guess is complex analysis uses things from real analysis. I'm doing very basic analysis in calc 2, and not sure if its enough to get by complex.
  9. B

    Proving the One-to-One Property and Image of a Complex Function

    Homework Statement Let f(z) = \frac{1-iz}{1+iz} and let \mathbb{D} = \{z : |z| < 1 \} . Prove that f is a one-to-one function and f(\mathbb{D}) = \{w : Re(w) > 0 \} . 2. The attempt at a solution I've already shown the first part: Assume f(z_1) = f(z_2) for some z_1, z_2 \in...
  10. J

    Easier to self-teach: differential geometry or complex analysis

    Hi all, I'm torn between taking complex analysis or differential geometry at the advanced third year level. Which of these would you consider the easiest to self-learn or the least applicable to the study of theoretical physics? I know that differential geometry shows up in general relativity...
  11. S

    Which text? First course in complex analysis

    Hi! I am signing up to take my first course in complex analysis this upcoming semester at my university. One of the professors with whom I am interested in taking the class is using Complex Analysis 2nd edition by Bak & Newman and the other one is using Complex Variables & Applications 7th...
  12. malawi_glenn

    Exp(tanz) = 1, complex analysis

    Homework Statement Find all solutions to: e^{\tan z} =1, z\in \mathbb{C}Homework Equations z = x+yi \log z = ln|z| + iargz +2\pihi, h\in \mathbb{Z}\log e^{z} = x + iy +2\pihi, h\in \mathbb{Z}Log e^{z} = x + iy The Attempt at a Solution I do not really know how to approach this, I tried to...
  13. malawi_glenn

    Complex cosine equation (complex analysis)

    Homework Statement Solve cosz = 2i , z\in \mathbb{C} The Attempt at a Solution e^{iz}+e^{-iz} = 4i t=e^{-z} t+t^{-1}=4i \Rightarrow t^{2}-4it+1=0 t = (2 \pm \sqrt{5})i log(e^{-z}) = logt z = x + yi;x,y \in \mathbb{R} log(e^{-z}) = log(e^{-y+ix}) = -y +xi +...
  14. T

    Advice: How do I master complex analysis in 5 weeks? ?

    Advice: How do I master complex analysis in 5 weeks? ??! Homework Statement Need to be throughly proficient with th first 7 chapters of saff and snider : fundamentals of complex analysis with engineering applications. Homework Equations egads! there's too many! The Attempt at a...
  15. L

    Is my proof correct for lim_(n-> infty) |z_n| = |z| ? Complex Analysis

    Is my proof correct for lim_(n-> infty) |z_n| = |z| ? Complex Analysis Homework Statement Show that if lim_{n-> infty} z_n = z then lim_{n-> infty} |z_n| = |z| Homework Equations The Attempt at a Solution Is this correct: lim_{n-> infty} |z_n| = |z| iff Assume...
  16. N

    Worldsheet and complex analysis stuff

    This really is a question on complex analysis but is about Polchinski's introduction to worlsdheet physics, so I am sure people here will answer this easily. I know it is a very basic question. Polchinski considers a field which is analytic and then says that because of this, one may write it...
  17. T

    Simple partial fractions help (warning complex analysis :P )

    Homework Statement the question can be ignored - it involves laplace and Z transforms of RLC ckts. Vc(s) = 0.2 ----------------- s^2 + 0.2s + 1 find the partial fraction equivalent such that it is : -j(0.1005) + j (0.1005) --------------...
  18. T

    Get Better at Complex Analysis: Urgent Help Needed

    I'm currently doing a course in complex analysis and we're using fundamentals of complex analysis, by saff and snider. https://www.amazon.com/dp/0139078746/?tag=pfamazon01-20 And Our problem sets are from the questions at the end of the chapters. I'm finding these questions incredibly hard...
  19. B

    Understanding Higher Order Poles in Conformal Transformations

    I suppose this is the proper place for this question:) I am learning about conformal field theories and have a question about poles of order > 1. If a conformal transformation acts as z \rightarrow f(z), f(z) must be both invertable and well-defined globally. I want to show that...
  20. C

    Dirichlet's Theorem (Complex Analysis): John B. Conway Explanation

    Hi could please let me know the Dirichlet's theorem(Complex analysis) ,statement atleast... as stated in John B Comway's book if possible ...I don't have the textbook and its urgent that's why...thank You
  21. A

    Understanding Poles and Zeros in Complex Analysis

    Oh god, so confused and panicked today:cry: I know this is a very basic question, but, givin the function 1/(z-w)^4 does this have one pole of order 4, or possibly 4 poles of order 1...? Also, could you please clarify, ''to get the zero's of a function, set the numerator = 0'' ''to...
  22. S

    Complex Analysis Textbook: Find the Best for Your Class

    Hello guys, Could someone reccomend me a complex analysis textbook? My class is currently using Stewart but i heard this is not such a good text...
  23. I

    Complex Analysis homework questions (some are challenging)

    1) Let U be a subset of C s.t U is open and connected and let f bea holomorphic function on U s.t. for every z in U, |f(z)| = 1, ie takes takes all points in U to the boundary of the unit circle. Prove that f is constant. Pf. Suppose f is not constant. Then we can find a w s.t. f'(w) is...
  24. M

    Complex Analysis Proof of Constant Function

    Homework Statement Prove that if a function f:c->c is analytic and lim as z to infinity of f(z)/z = 0 that f is constant. Homework Equations Cauchy Integral Formula for the first derivative (want to show this is 0 ie: constant) f prime (z) = 1/(2ipi)*Integral over alpha (circle radius...
  25. B

    Solving Complex Analysis Problems - Get Advice Here!

    Can anyone give me some advice on how to solve this problem? in the reflection principle if f(x) is pure imaginary then the conjugate of f(z)=-f(z*) where z* is the complex conjugate of z. Any advice on where to start? thanks
  26. B

    Solving Complex Analysis Problems: Where to Start?

    Can anyone give me some advice on how to solve this problem? in the reflection principle if f(x) is pure imaginary then the conjugate of f(z)=-f(z*) where z* is the complex conjugate of z. Any advice on where to start? thanks
  27. I

    Residue Calculation for Complex Analysis - Exercise Solution Discrepancy

    Hello! I am studing for my Complex Analysis exam and solving the exercises for Residues given by the professor. The problem is that for some exercises I get to a solution different from the one of the professor :bugeye:, and I am not sure that the mistake is in my calculations. I would...
  28. A

    Evaluating Integrals Using Residues and Proving Complex Analysis Equations

    Homework Statement 1. Evaluate the following integrals using residues: a) \int _0 ^{\infty} \frac{x^{1/4}}{1 + x^3}dx b) \int _{-\infty} ^{\infty} \frac{\cos (x)}{1 + x^4}dx c) \int _0 ^{\infty} \frac{dx}{p(x)} where p(x) is a poly. with no zeros on {x > 0} d) \int _{-\infty}...
  29. A

    Complex Analysis Homework: Show Polynomial Degree ≤ n, Is f Polynomial?

    Homework Statement 1. Suppose that f(z) is holomorphic in C and that |f(z)| < M|z|n for |z| > R, where M, R > 0. Show that f(z) is a polynomial of degree at most n. 2. Let f(z) be a holomorphic function on a disk |z| < r and suppose that f(z)2 is a polynomial. Is f(z) a polynomial? Why...
  30. G

    A series for sin az / sin pi z in complex analysis

    Homework Statement Show that \frac{\sin (az)}{\sin (\pi z)} = \frac{2}{\pi} \sum_{n=1}^{+\infty} (-1)^n \frac{n \sin (an)}{z^2 - n^2} for all a such that - \pi < a < \pi Homework Equations None really, we have similar expansions for \pi cot (\pi z) and \pi / \sin (\pi z) , this...
  31. R

    Complex Analysis: Calculating the Limit of I(r)

    Some hints/help woudl be greatly appreciated! Let I(r) = integral over gamma of (e^iz)/z where gamma: [0,pi] -> C is defined by gamma(t) = re^it. Show that lim r -> infinity of I(r) = 0.
  32. B

    Complex analysis and delta epsilon proof

    Can anyone help with these problems 1). use def. delta epsilon proof to prove lim(z goes to z0) Re(z)=Re(z0) This is what I did |Re(z)-Re(z0)| = |x-x0| < epsilon then |z-z0|=|x-iy-x0-iy0|=|x-x0+i(y-y0)|<=|x-x0|+|y-y0|=epsilon + |y-y0| = delta My question is doesn't this delta have to...
  33. D

    I need super help on a complex analysis problem.

    Homework Statement Is there a Laurent Series for Log(z) in the Annulus 0<|z|<1? Homework Equations Go here for the Theorem. It is theorem 7.8: www.math.fullerton.edu/mathews/c2003/LaurentSeriesMod.html[/URL] (copy and paste the link below if you are having problems. Exclude the "[url]" in...
  34. J

    Book recommendations - complex analysis

    I need a book that's semi-introductory (advanced undergrad to beginning graduate level, if possible) on complex analysis, particularly one that covers power series well, but should be fairly general. I currently have "elementary real and complex analysis" by Georgi Shilov and while it's not...
  35. T

    Interesting Idea: Showing a Force is Conservative with Complex Analysis

    Preface:The best way I've been taught how to prove that a force is conservative is to take the curl of the force and show that it is equal to zero. That's pretty quick, but after studying for a complex analysis midterm this idea struck my mind. I'm not a master of complex analysis, so there...
  36. I

    Complex Analysis: Finding Arg(z)

    Hello everyone, I am trying to solve this follow problem, but don't quite know how to go about getting Arg(z). z = 6 / (1 + 4i) I got that lzl is sqrt((6/17)^2+(-24/17)^2) but am stuck with finding Arg(z). It told me to recall that -pi < Arg(z) <= pi Can you guys teach me how to go...
  37. P

    Complex Analysis question

    So my professor threw in what he called an extra 'hard' question for a practice test. So naturally I have a question about it. It relates to the Maximum Modulus Principle: a) Let p(z) = a_0 + a_1 z + a_2 z^2 + ... and let M = max |p(z)| on |z|=1. Show that |a_i|< M for i = 0,1,2. b)...
  38. G

    What are the Applications of Complex Analysis in Calculus?

    for our project in calculus, I am doing a presentation on the basics of complex analysis. Somewhere along there I need to tackle the question: what are the applications of complex analysis? Are there any application problems that I can give that involve basic derivatives/integrals of complex...
  39. P

    Elementary Real and Complex Analysis

    Hi. So I was reading through "Elementary Real and Complex Analysis" by Georgi E. Shilov (reading the first chapter on Real Numbers and all that "simple" stuff like the field axioms, a bit of set stuff, etc.). Anyways, so while I was reading, I ran into something I couldn't understand... the...
  40. J

    Solving Complex Integration: Principal Value and Summation with Contour Methods

    I have two questions on complex integration, and I do not know how to solve them. Please help if you can. Thanks 1. Evaluate the following principal value integral using an appropriate contour. Integration of (integral goes from 0 to infinity) : (x)^a-1/1-x^2, 0<a<1. 2.Using contour...
  41. S

    Complex analysis - area inside a simple closed curve

    Let C be a simple closed curve. Show that the area enclosed by C is given by 1/2i * integral of conjugate of z over the curve C with respect to z. the hint says: use polar coordinates i can prove it for a circle, but i am not sure how to extend it to prove it for any given closed curve
  42. S

    Complex analysis - maximum/modulus principle

    Suppose that f is analytic on a domain D, which contains a simple closed curve lambda and the inside of lambda. If |f| is constant on lambda, then either f is constant or f has a zero inside lambda ... i am supposed to use maximum/modulus principle to prove it ... here is my take: if f...
  43. S

    Complex analysis - LFT

    Verify that the linear fractional transformation T(z) = (z2 - z1) / (z - z1) maps z1 to infinity, z2 to 1 and infinity to zero. ^^^ so for problems like these, do I just plug in z1, z2 and infinity in the eqn given for T(z) and see what value they give? in this case, do i assume 1/ 0 is...
  44. S

    Complex analysis - argument principle

    (changes in arg h (z) as z traverses lambda)/(2pi) = # of zeroes of h inside lambda + # of holes of h inside lambda now the doubt i have is what happens when the change i get in h (z) is say 9 pi/2 ... because then i would have a 2.5 on left side of the eqn ... so do i round it up and...
  45. S

    Complex analysis - conformal maps -mapping

    find a one-to-one analytic function that maps the domain {} to upper half plane etc ... for questions like these, do we just have to be blessed with good intuition or there are actually sound mathematical ways to come up with one-to-one analytic functions that satisfy the given requirement...
  46. E

    Complex analysis - Cauchy Theorem

    Hi again. Can somebody help me out with this question? "\int_{C_1(0)} \frac {e^{z^n + z^{n-1}+...+ z + 1}} {e^{z^2}} \,dz Where C_r(p) is a circle with centre p and radius r, traced anticlockwise." I'd be guessing that you have to compare this integral with the Cauchy integral formula...
  47. E

    Another Complex Analysis Question

    Suppose you have a Meromorphic function f(z) that has a zero at some point in the complex plane. Suppose you create two parallel contours Y1 and Y2 that are parallel and infinitely close to each other yet still contains the zero (the contours are infinitely close to the zero but don't run...
  48. S

    Harmonic functions - complex analysis

    so .. if f (z) = u + iv is analytic on D, then u and v are harmonic on D... now ... if f (z) never vanishes on the domain ... then show log |f (z)| is harmonic on the domain ... Recall: harmonic means second partial derivative of f with respect to x + second partial derivative of f with...
  49. E

    Complex Analysis and Change of Variables in Line Integrals

    Consider the function: g(z(t)) = i*f '(c+it)/(f(c+it) - a) Where {-d <= t < d} If we let z = c+it By change of variables don't we get: Line integral of g(z(t)) = i ln[f(c+it) - a] evaluated from t = - d to t = d? note: ln is the natural log. Inquisitively, Edwin...
  50. E

    Complex analysis taylor series Q

    hi, I'm wondering if someone can help me out with this question: "What are the first two non-zero terms of the Taylor series f(z) = \frac {sin(z)} {1 - z^4} expanded about z = 0. (Don't use any differentiation. Just cross multiply and do mental arithmetic)" I know the formula for...
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