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. S

    Complex analysis - something really confusing

    I think I have misunderstood one of the theorems in complex analysis (k reperesents the order of the derivative) Theorem: Suppose f is analytic on a domain D and, further, at some point z0 subset of D, f (k) (z0) = 0. Then f(z) = 0 for all z subset of D ... Is the theorem basically...
  2. W

    Proving Equality of Analytic Functions on a Simple Loop

    Here's my question: Let f and g be analytic inside and on the smple loop \Gamma. Prove that if f(z)=g(z) for all z on \Gamma, then f(z)=g(z) for all z inside \Gamma. Don't really know where to start on this one. This comes from the section 'Cauchy's Integral Formula'.
  3. E

    Can Complex Analysis Techniques Split Double Poles into Isolated Singularities?

    I'm glad to see that the physics forum website is back online. Suppose you have a function with double poles somewhere on the complex plane. Are there complex analysis techniques that can be used to split the double pole into two single isolated poles? Some example functions might be...
  4. cepheid

    Complex Analysis => Fluid Flow

    I'm struggling with this question right now: Let the complex velocity potential \Omega(z) be defined implicitly by z = \Omega + e^{\Omega} Show that this map corresponds to (some kind of fluid flow, shown in a diagram, not important). For background, \Omega = \Phi + i\Psi...
  5. quasar987

    Complex Analysis: Evaluating Integrals with Contractible Jordan Paths

    Consider a domain D and f:D-->\mathbb{C} a holomorphic function and C a contractible Jordan path contained in D and z1, z2, two points in the interior of C. Evaluate \int_C \frac{f(z)}{(z-z_1)(z-z_2)}dz What happens as z_1 \rightarrow z_2? I have found that \int_C...
  6. P

    How can I prove that if wz = 0, then w = 0 or z = 0 in complex analysis?

    Prove that if wz = 0, then w = 0 or z = 0. w and z are two complex numbers. I said that w = a + bi and z = c + di and set wz = 0. I got down to c(a+b) = d(b-a), but don't know where to go from here. I'm trying to teach myself complex analysis, anyone know any good sources? I took the 3...
  7. J

    Complex Analysis: Defining Complex Volume & Sphere w/ Winding Number

    Suppose you have a unit circle in the complex plane e^{it}, -\infty \leq t \leq \infty. The contour will wind around forever, so at all points in the contour, there are an infinite amount of possible winding numbers, although they are all multiples of 2pi with a well defined contour boundry...
  8. F

    Integrating Sin(1/z) and Z Sin (1/Z^3) Over C (Circle of Radius 1)

    hi there im confused with this question.. Integrate 1) Sin(1/z) dz and 2) Z sin (1/Z^3) where Z is any complex number., over C which is a circle of radius 1 centred at 0 i tried using the cauchy integral formula and stuff but somehow the answer always comes infinity...is...
  9. T

    Solving a Complex Analysis Problem: Finding Critical Points of k(x)

    "Let a,b be in R with a>0 and f(x)=ax^3+bx. Let k(x)=[f''(x)]/[1+(f'(x))^2]^(3/2). Find the critical points of k(x) and use the first derivative test to classify them." This seems incredibly quantitative and complicated for an analysis assignment. There must be a theorem of some kind I can...
  10. D

    Complex Analysis: Find Laurent Series for f(z) = 1/(e^-z - 1) About 0

    Hi there, I'm taking this math for physicists course and we're doing some stuff with functions of complex variables (laurent series residue etc), and I"m having a bit of trouble. I'm not so happy with the book we use. It's a great reference book if you know what you're doing already but...
  11. N

    Explicit Use of Complex Analysis in Physics: What Courses?

    Can someone tell me when explicit use of complex analysis arises in physics? What particular courses? Thanks
  12. S

    Complex analysis- Oscillation/vibration class

    complex analysis-- Oscillation/vibration class Hello all, I'm taking a wave/vibration/oscillation class, and we're delving into complex notation for these. One of our assigments dealt with a complex function that we didn't get a whole lot of practice out of in math methods. I've gone back...
  13. T

    How can complex analysis be applied to Einstein's theory of relativity?

    As you know complex analysis has provided many useful tools for harmonic analysis. However I think its application to Einstein's theory of relativity is relatively limited. So I tried to modify complex analysis in order to apply it to the theory of relativity more easily in the following...
  14. T

    What is the process for composing rotations in Visual Complex Analysis?

    "Visual Complex Analysis" I have gotten myself wound around the axel regarding something in "Visual Complex Analysis" (Dr. Tristan Needham) that should be easy. On p. 18 (paperback edition), towards the bottom, the result for two rotations about different points has got me stumped. I cannot...
  15. S

    Is Cauchy's Theorem the Best Starting Point for Learning Complex Analysis?

    hello all well i am going to slowly research my way into complex analysis and I decided to start with cauchys theorem i hope this is the best part to start with, well anyway it says that if f(z) is analytic and \frac{f(z)}{z-z_{o}} has a simple pole at z_{0} with residue f(z_{o}) then...
  16. H

    Importance of Complex Analysis

    Hi PPls okay i have studied calculus and i can easily see its application in many things like calculating volume,areas,rates ..etc. but i want to know what is the application of complex analysis...where does it all find its uses and why one study it??
  17. H

    No Complex Analysis Equation z^4 + z + 5 = 0

    Hi all,, I have a problems on complex Analysis: Show that the equation z^4 + z + 5 = 0 has no solution in the set { z is a subset of C: modulus of z is less than 1} i tried doing it using Triangle inequality although i got it but i am looking for a better solution...Pls help
  18. R

    Conformal mapping in Complex Analysis

    I would appreciate if someone could explain Conformal Mapping using Complex Analysis using an example. I get the rough idea but have no clue how complex analysis comes into the picture. Thank You!
  19. P

    Complex Analysis Question

    This one is pretty involved so mad props to whoever can help me figure it out. I've been thinking about this for more than an hour and it's bugging me. Consider the function f(z) = 1/(sin(pi/z)). It has singular points at z=0 and z=1/n (where n is an integer). However, my book says each...
  20. G

    Complex Analysis: Applications & Virtual Classrooms

    I was just wondering what exactly complex analysis is, and what type of applications it's study can be applied to. By the way, an excellent discussion/class on differential forms is taking place here , if anyone would be interested in starting a similar type forum on complex analysis, that...
  21. N

    Need some help with basic complex analysis (no proofs)

    need some urgent help with basic complex analysis (no proofs) This forum is probably more appropriate. please forgive me for double posting. Can someone give me examples of the following? (no proofs needed) (C is the complex set) 1. a non-zero complex number z such that Arg(z^2) is NOT...
  22. F

    Complex analysis question (only need a hint)

    i know it's supposed to be a simple question. frustrating because it is not coming to me. just want a hint. question is: how do you write 1 + cos(theta) + cos (2*theta) + cos(3*theta)... cos(n*theta) using the fact that (z^(n+1) -1) / (z^(n) -1) = 1 + z + z^(2) +... + z^(n) thanks in...
  23. R

    How to Evaluate a Cauchy Integral on Different Paths?

    Cauchy integral question The question calls for finding the integral of dz/((z-i)(z+1)) (C:|z-i|=1) I can't figure out how to do this for (C:|z-i|=1). How does this differ from, say, (C: |z|=2) Regards
  24. A

    Complex Analysis: Nonlinear system

    Here's a problem I ran into in complex analysis. Given z = x + iy and w = u + iv, I need to find all w such that w² = z. It reduces to solving this system: x = u² - v² y = 2uv My professor mentioned that we should try to deal with the problem in at least two cases: y = 0, and y does not...
  25. J

    Introduction to Complex Analysis

    Hi people, I'm Joseph, 17, English studying European Baccalaureate. I was wondering if anyone here could recommend for me a good introductory book on Complex Analysis that requires only an understanding of the complex numbers you would cover in High School Maths. Maybe something...
  26. Y

    Complex Analysis finding a sum

    Anyone know how to find a sum of a function using the poisson summation formula and the Fourier transform. Thanks! --yxgao
  27. P

    More Complex Complex Analysis

    More "Complex" Complex Analysis I have another problem that has eluded me for days and I'm sure I'm close. If anyone can help, please nudge me in the right direction. Consider the mapping w = u + iv = 1/z, where z = x + iy. Show that the region between the curves v = -1 and v = 0 maps into...
  28. P

    What is the derivative of a complex function?

    I am having trouble with the following question, any help would be blinding. Find the value of ther derivative of: (z - i)/(z + i) at i. I tried to use the fact that f'(z0) = lim z->z0 [f(z) - f(z0)]/z - z0. I also tried using the fact that z = x + iy and rationalising the denominator...
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