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. M

    Complex Analysis: Entire function dominated by another entire function

    Homework Statement If f,g are entire functions and |f(z)| <= |g(z)| for all z, draw some conclusions about the relationship between f and g Homework Equations none The Attempt at a Solution I just need a push in the right direction.. thanks for any and all help!
  2. G

    Complex Analysis: Proving a function is equivalent to its series representation

    Homework Statement Compare the function f(z) = (pi/sin(pi*z))^2 to the summation of g(z) = 1/(z-n)^2 for n ranging from negative infinity to infinity. Show that their difference is 1) pole-free, i.e. analytic 2) of period 1 3) bounded in the strip 0 < x < 1 Conclude that they are...
  3. S

    Basic Complex Analysis: Maximum Modulus?

    Homework Statement Let f and g be two holomorphic functions in the unit disc D1 = {z : |z| < 1}, continuous in D1, which do not vanish for any value of z in the closure of D1. Assume that |f(z)| = |g(z)| for every z in the boundary of D1 and moreover f(1) = g(1). Prove that f and g are the...
  4. S

    Basic Complex Analysis: Uniform convergence of derivatives to 0

    Homework Statement Let f_n be a sequence of holomorphic functions such that f_n converges to zero uniformly in the disc D1 = {z : |z| < 1}. Prove that f '_n converges to zero uniformly in D = {z : |z| < 1/2}.Homework Equations Cauchy inequalities (estimates from the Cauchy integral formula)The...
  5. S

    Basic Complex Analysis: Cauchy Riemann

    Homework Statement Let f be a holomorphic function in the unit disc D1 whose real part is constant. Prove that the imaginary part is also constant. Homework Equations Cauchy Riemann equations The Attempt at a Solution Hi guys, I'm working through the basics again. I think here we...
  6. G

    Complex analysis: mapping a hyperbola onto a line

    Homework Statement We want to create a map from (x,y) to (u,v) such that the right side (positive x) of the hyperbola x^2 - y^2 = 1 is mapped onto the line v = 0 AND all the points to the left of that hyperbola are mapped to above the line. The mapping should be one-to-one and conformal...
  7. T

    Complex Analysis: Maximum Modulus Principle

    Homework Statement Let f(z) = u(x,y) + iv(x,y) be a continuous, non-constant function that is analytic on some closed bounded region R. Prove that the component function u(x,y) reaches a minimum value on the boundary of R. The Attempt at a Solution By the minimum modulus principle...
  8. T

    Simple Complex Analysis Question

    Homework Statement Suppose f(z) is entire and the harmonic function u(x,y) = Re[f(z)] has an upper bound u_0. (i.e. u(x,y) <= u_0 for all real numbers x and y). Show that u(x,y) must be constant throughout the plane. The Attempt at a Solution Since f(z) = u(x,y) + iv(x,y) is entire...
  9. MathWarrior

    Math classes to become good at complex analysis?

    What undergraduate math classes would you want to take if you wanted to be exposed to the stuff used in complex analysis? (Besides complex analysis)
  10. G

    Integration using complex analysis

    I have to integrate S cos^8 (t) dt from 0 to 2 pi, presumably using complex analysis I got to S [(e^(it) + e(-it))/2]^8 dt from 0 ti 2pi How do I take it from here? I have a hint- use binomial theorem.
  11. M

    Proving Inequalities for Complex Analysis Limits

    Homework Statement I'm not very good with LaTeX and the reference button seems to broken. So Assume lim h(z) = 1+i, as z->w, prove there exists a delta, d>0 s.t. 0<|z-w|<d -> (2^.5)/2 < |h(z)| < 3(2^.5)/2 Homework Equations The Attempt at a Solution Kinda been running in...
  12. H

    Complex Analysis Complex Integration Question

    Its question 1(g) in the picture. My work is shown there as well. This has to do with independence of path of a contour. Reason I am suspicious is that first there is a different answer online and second it says "principal branch" which I have not understood. Does that mean a straight line for...
  13. S

    Why Are Branch Cuts Necessary in Complex Analysis?

    I was hoping someone could clarify the idea of a branch cut for me. In class, my professor talked about how a branch cut is used to remove discontinuities. He gave an example of |z|=1 needing a branch cut along the positive real axis. If this because going from 0 to 2\pi, the 0 and 2\pi match up?
  14. T

    Proving Entirety of conj(f(conj(z))) for an Entire Function f

    Homework Statement Show that if a function f(z) = u(x,y) +iv(x,y) is entire, then the function conj(f(conj(z))) is entire. Homework Equations (i) The Cauchy-Riemann (CR) equations hold for functions that are entire: u_x = v_y and u_y = -v_x (ii) conj(_) is the conjugate (i.e. there...
  15. T

    Complex Analysis Concept Question

    Homework Statement Just to make certain that I am understanding this correctly, given a function f(z) = u(x,y) + iv(x,y), the existence of the satisfaction of the Cauchy-Riemann equations alone does not guarantee differentiability, but if those partial derivatives are continuous and the...
  16. T

    How to Show u(x,y) and v(x,y) are Constant Throughout D?

    Homework Statement Suppose v is a harmonic conjugate of u in a domain D, and that u is a harmonic conjugate of v in D. Show how it follows that u(x,y) and v(x,y) are constant throughout D. The Attempt at a Solution since u is a harmonic conjugate of v, u_xx + u_yy = 0 also, since v...
  17. P

    Complex Analysis: Express f(z)= (z+i)/(z^2+1) as w=u(x,y)+iv(x,y)

    Homework Statement write f(z)= (z+i)/(z^2+1) in the form w=u(x,y)+iv(x,y) Homework Equations The Attempt at a Solution I tried using the conjugate and also expanding out algebraically but I can not seem to get the right answer. I know what the answer is...
  18. T

    Is the Function g(z) = ln(r) + i(theta) Analytic and What Are Its Derivatives?

    Homework Statement (a) Use the polar form of the Cauchy-Riemann equations to show that: g(z) = ln(r) + i(theta); r > 0 and 0 < (theta) < 2pi is analytic in the given region and find its derivative. (b) then show that the composite function G(z) = g(z^2 + 1) is analytic in the...
  19. G

    Radius of Converge Complex Analysis

    Homework Statement Find the radius of convergence of \sumcnz^{n} if c2k = 2^{k} and c2k-1 = (1+1/k)^{k^{2}}, k = 1, 2, 3... Homework Equations 1/R = limsup as n=> infinity |cn|^1/nThe Attempt at a Solution I'm not really sure where to start with this. I know that it's a power series, and to...
  20. E

    Why use a laurent series in complex analysis?

    In complex analysis, what exactly is the purpuse of the luarent series, i mean, i know that it apporximates the function like a taylor series, an if the function is analytic in the whole domain it simplifies into a taylor series. But i fail to see its purpose - what does it do that the taylor...
  21. G

    Complex Analysis: Radius of Convergence

    Homework Statement Find the radius of convergence of the power series: a) \sum z^{n!} n=0 to infinity b) \sum (n+2^{n})z^{n} n=0 to infinity Homework Equations Radius = 1/(limsup n=>infinity |cn|^1/n) The Attempt at a Solution a) Is cn in this case just 1? And plugging it in...
  22. J

    Complex Analysis: polynomial coefficients

    Homework Statement Show that if the polynomial p(z)=anzn+an-1zn-1+...+a0 is written in factored form as p(z)=an(z-z1)d1(z-z2)d2...(z-zr)dr, then (a) n=d1+d2+...+dr (b) an-1=-an(d1z1+d2z2+...+drzr) (c) a0=an(-1)rz1d1z2d2...zrdr Homework Equations Taylor form of a polynomial? p(z) = \sum...
  23. R

    Complex analysis antiderivative existence

    Homework Statement a) Does f(z)=1/z have an antiderivative over C/(0,0)? b) Does f(z)=(1/z)^n have an antiderivative over C/(0,0), n integer and not equal to 1. Homework Equations The Attempt at a Solution a) No. Integrating over C= the unit circle gives us 2*pi*i. So for at least one...
  24. T

    Did I do this complex analysis proof right?

    Homework Statement Show that if c is any nth root of unity other than unity itself that: 1 + c + c^2 + ... + c^(n-1) = 0 Homework Equations 1 + z + z^2 + ... + z^n = (1 - z^(n+1)) / (1 - z) The Attempt at a Solution c is an nth root of unity other than unity itself => (1-c) =/= 0. so, 1 + c...
  25. T

    Master Complex Analysis: Homework Statement, Equations, and Solutions

    Homework Statement Homework Equations The Attempt at a Solution How do I go about Q1 and showing the coefficients are unique and then Q2?
  26. M

    Residues for a Complex Analysis Noob

    I need to calculate the residue of ( 1 - cos wt ) / w^2 This has a pole of second order at w=0, am I correct? Now may math book says that a second order residue is given by limit z goes to z_0 of {[(z-z_0)^2. f(z)]'} where z_0 is the pole I'm quite new to complex...
  27. S

    Complex Analysis - Proving a bijection on a closed disk

    Homework Statement For each w \in \mathbb{C} define the function \phi_w on the open set \mathbb{C}\backslash \{\bar{w}^{-1}\} by \phi_w (z) = \frac{w - z}{1 - \bar{w}z}, for z \in \mathbb{C}\backslash \{\bar{w}^{-1}\} \back. Prove that \phi_w : \bar{D} \mapsto \bar{D} is a...
  28. R

    Using the Residue Theorem for Complex Analysis Integrals

    Homework Statement Use the residue theorem to compute \int_0^{2\pi} sin^{2n}\theta\ d\theta Homework Equations \mathrm{Res}(f,c) = \frac{1}{(n-1)!} \lim_{z \to c} \frac{d^{n-1}}{dz^{n-1}}\left( (z-c)^{n}f(z) \right) The Attempt at a Solution I started with the substitution z = e^{i\theta}...
  29. R

    'constant' functions on complex analysis

    Okay, so, I don't understand this concept of 'maximum principle'. A few weeks ago we did Liouville's theorem, which states that any bounded complex function is continuous. Okay... (I can't really imagine the picture of a function which is bounded to be constant, e.g. sin(z) is bounded, at...
  30. K

    Should be a basic complex analysis question

    Homework Statement Let f:C-> C be an entire bijection with a never zero derivative, then f(z)=az+b for a,b\in CHomework Equations The Attempt at a Solution I'm not sure where to begin with this problem. The only ways I see to attack this are based on somehow showing that f' is bounded and then...
  31. I

    Trigonometric integral / Complex Analysis

    Homework Statement Calculate the integral \int\limits_0^{\infty} \cos{x^2} dx This is the exercise from complex analysis chapter, so I guess I should change it into a complex integral somehow and than integrate. I just don't know how, since neither substitution cos(x^2) = Re{e^ix^2} nor...
  32. S

    Suggested Textbooks for Complex Analysis w/Proofs & Accessible w/o Real Analysis

    I don't think this is the right area to post this question so to the mods: please be kind and move it to a better section if one exists. I'm looking for a textbook on complex analysis which gives proofs but is accessible without a formal real analysis course. I would appreciate suggestions...
  33. C

    Getting Residues Complex Analysis

    Hi everyone! I still didn't fully understand how to get the residues of a complex function. For example the function f(z)=\frac{1}{(z^{2}-1)^{2}} in the region 0<|z-1|<2 has a pole of order 2. So the residue of f(z) in 1 should be given by the limit: \lim_{z \to 1}(z-1)^{2}f(z)=1/4 But...
  34. P

    Complex analysis, integral independent of path

    Homework Statement when complex integral is independent of path? i heard that its for every function f(z) but when i have function f(z)=\left(x^2+y\right)+i\left(xy\right) its not independent, why?
  35. P

    Complex analysis - integral independent of path

    Homework Statement integral: \int\limits_C\cos\frac{z}{2}\mbox{d}z where C is any curve from 0 to \pi+2i The Attempt at a Solution can i do this like in real analysis when counting work between two points, just count this integral and put given data in?
  36. P

    Analyzing a Complex Integral: Circle of Radius 2 Centered at 0 Counterclockwise

    Homework Statement integral: \int\limits_C\frac{\mbox{d}z}{z} where C is circle of radius 2 centered at 0 oriented counterclockwise Homework Equations The Attempt at a Solution I am going to parameter this: \gamma=2\cos t+2i\sin t,\ \gamma^\prime=-2\sin t+2i\cos t,\ t\in[0,2\pi], then...
  37. B

    How Do You Solve Part (b) for a Bounded |f''(z)| in a Maclaurin Series Problem?

    Suppose that f is entire,= and that f(0)=f'(0)=f''(0)=1 (a) Write the first three terms of the Maclaurin series for f(z) (b) Suppose also that |f''(z)| is bounded. Find a formula for f(z). I believe (a) is just 1+z+(z^2)/2! however (b) I do not know where to begin.
  38. B

    Complex Analysis Singularities and Poles

    Assume throughout that f is analytic, with a zero of order 42 at z=0. (a)What kind of zero does f' have at z=0? Why? (b)What kind of singularity does 1/f have at z=0? Why? (c)What kind of singularity does f'/f have at z=0? Why? for (a) I'm pretty sure it is a zero of order 41...
  39. B

    Complex Analysis Entire Functions

    Let f(z) be an entire function such that |f(z)| less that or equal to R whenever R>0 and |z|=R. (a)Show that f''(0)=0=f'''(0)=f''''(0)=... (b)Show that f(0)=0. (c) Give two example of such a function f.
  40. B

    Complex Analysis and Analytic Functions

    Let f be analytic for |z| less than or equal to 1 and suppose that |f(z)| less than or equal to |e^z| when |z|=1. Show (a)|f(z)| less than or equal to |e^z| when |z|<1 and (b)If f(0)=i, then f(z)=ie^z for all z with |z| less than or equal to 1
  41. C

    Complex Analysis (Argument Principle to determine location of roots)

    Homework Statement With f(z) = 2z^{4} +2z^{3} +z^{2} +8z +1 Show that f has exactly one zero in the open first quadrant.Homework Equations Argument PrincipleThe Attempt at a Solution I know I'm supposed to use the Argument Principle.. So far, all I can do is show something like, in the unit...
  42. D

    Which class should I take next semester, Complex Analysis or Topology?

    Hi, I'm a junior undergrad majoring in math and physics, and am deciding between complex analysis and topology for next semester. (I'm planning on doing theoretical physics for grad, something on the more mathematical side, so topology would likely be used). Complex Analysis Pros...
  43. F

    How can e^(1/z) be written using the definition of e^z?

    How do you write e^(1/z) in the other form? z = x+yi So we should be able to right it using this definition of e^z, no? e^z = e^x * [cos(y) + i * sin(y)] I pushed some numbers around the page for a while but I can't get 1/(x+i*y) to split into anything nice. Is there a trick?
  44. C

    Complex Analysis (zeroes of Polynomials)

    I just wanted to know what kind of math is needed to solve questions like 1, 2 and 3 of http://www.math.toronto.edu/deljunco/354/ps4.fall10.pdf and number 5 of http://www.math.toronto.edu/deljunco/354/354final08.pdf . I don't need solutions, I just need to know what book or online source can...
  45. Z

    Wave Formula by complex analysis

    How can I express the general wave formula, y=Acos(wt-kx), by the complex ft and the exponential ft? Is it right to use Euler's Identity?
  46. murshid_islam

    Please suggest a good book on Complex Analysis

    Can anyone suggest a good book on Complex Analysis? I need a book that would be good for self studying.
  47. B

    Proving Analytic Function Bounds: Complex Analysis Help and Tips

    Suppose f is analytic inside |z|=1. Prove that if |f(z)| is less than or equal to M for |z|=1, then |f(0)| is less than or equal M and |f'(0)| is less than or equal to M. I'm really stuck here on how to approach this problem. Help PLZ!
  48. B

    Proving Analyticity of u_x - iu_y in Complex Analysis

    Suppose that u(x,y) is harmonic for all (x,y). Show that u_x-iu_y is analytic for all z. (Assume that all derivatives in the question exist and are continuous) I have no idea where to start with this? Something with the Cauchy Riemann equations is required but I'm not sure exactly how to...
  49. R

    Solving Complex Analysis Problem: Calculating Index of a Curve

    Homework Statement This is complex analysis by the way. Here's the problem statement:http://i.imgur.com/wegWj.png" I'm doing part b, but some information from part a is carried over. The Attempt at a Solution My problem is that I don't know if I'm being asked to show it via direct...
  50. L

    Stolz Angle and Complex Analysis

    Homework Statement Show that Abel's Limit Theorem holds as the complex number z approaches 1 if instead of taking the requirement |1 - z| \leq M|1-|z|| , you restrict z to |arg(1- z)| \leq \alpha where 0 < \alpha < \frac{\pi}{2} Homework Equations The Attempt at a Solution The latter...
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