What is Complex function: Definition and 140 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. F

    Determine where a complex function is analytic

    Homework Statement Determine the region of analyticity of ln|z| + i*arg(z) and justify your answer. Homework Equations The Attempt at a Solution I said that if z=x+iy, the function has a singularity when both x and y are equal to 0 since ln(0) is undefined and arg(z) = arg(x+iy) =...
  2. R

    Limiting a Complex Function to π

    {{\lim_{\substack{x\rightarrow\pi}} {\left( \frac {x}{x-\pi}{\int_{\pi}^{x} }\frac{sin t}{t}} dt\right)}
  3. J

    Differentiability of a complex function.

    (PROBLEM SOLVED) I am trying to think of a complex function that is nowhere differentiable except at the origin and on the circle of radius 1, centered at the origin. I have tried using the Cauchy-Riemann equations (where f(x+iy)=u(x,y)+iv(x,y)) \frac{\partial u}{\partial x}=\frac{\partial...
  4. H

    Maxima, Minima complex function

    Hey my problem is that I am unable to calculate the absolute value of the following function: f(z)=\bar{z}(z-2)-2\Re z wherase z=x+iy What i did was: =|z|^2-2\bar{z}-2\Re z=x^2+y^2-2x+2iy-2x=x^2+y^2+2yi-4x and how should i calculate the absoulte value of this function?? Because i should...
  5. S

    What Are the Roots of z^n = -1 in Complex Numbers?

    Homework Statement What are the roots of z^n = -1 Homework Equations The Attempt at a Solution are they e^{\frac{2\pi k i}{n}-\frac{i \pi}{n}} ?
  6. M

    Finding the Real Points of a Complex Function

    Hi all, I'm trying to solve the following expression for w when w is entirely real (no imaginary component). It is part of a circuits problem where I have to design a band pass filter. Note that I'm using j instead of i to denote the imaginary component...
  7. C

    How Do You Prove the Limit of a Complex Function Using Epsilon-Delta?

    Homework Statement Use epsilon-delta proof to show that \lim_{z\to z_0}(z^2+c)=z_0^2+c. Homework Equations \forall\epsilon>0 \exists\delta>0 \forall z (|z-z_0|<\delta\Rightarrow|f(z)-\omega_0|<\epsilon) The Attempt at a Solution So f(z)=z^2+c and \omega_0=z_0^2+c. In order to write my...
  8. T

    How do you show that a complex function is analytical?

    Like if I wanted to show how sin z, cos z, or e^z are analytical, what is the general process I have to do? Can I use the cauchy - riemann relations somehow? (where z = x + iy is complex)
  9. L

    STRACT: Understanding the Complex Conjugate of a Function

    Homework Statement I have a complex function w\left(z\right)=e^{sin\left(z\right)} What is the conjugate? 2. The attempt at a solution The conjugate is w\left(z^{*}\right)=e^{sin\left(z^{*}\right)} w\left(x-iy\right)=e^{sin\left(x-iy\right)} My question is, is my answer...
  10. L

    Mapping of multivalued complex function.

    A complex function f\left(z\right)=\sqrt{z} can be splitted into two branches: 1. Principal branch: f_{1}\left(z\right)=\sqrt{r} e^{i \left(\theta/2\right)} 2. Second branch: f_{2}\left(z\right)=\sqrt{r} e^{i \left[\left(\theta+2\pi\right) /2\right]} My question is, is there a way to...
  11. J

    Principal Part of Complex Function

    I'm trying to do a question about finding the principal part of a complex valued function: f(z)= \frac{1}{(1+z^3)^2} . I really don't know how to go about even starting it, any tips? (just in case my terminology is different to anyone elses, the principal part is the terms of the Laurent...
  12. V

    Analyzing the Analyticity of ln (z)

    Homework Statement How would I prove that ln (z) is analytic? Homework Equations ... The Attempt at a Solution I rewrote it as ln (z) = ln (r) + i\theta. But, I'm not quite sure how to apply Cauchy-Riemman conditions here.
  13. P

    Power series for complex function

    Homework Statement Find the power series for the function f(z) = (1-z)^-m Hint: Differentiation gives: f'(z) = m(1-z)^m-1 = m(1-z)^-1.f(z) or: zf'(z) + mf(z) = f'(z) Use the formula for differentiation of power series to determine the coefficients of the power series for f...
  14. O

    How Is the Color Domain Used to Graph Complex Functions in Polar Coordinates?

    As you know to graph a complex function we need four dimensional system,but i encountered with some graphs of complex functions on polar coordinate systems which called "colour domain". Can somebody explain me what is the colour domain method and how to graph a complex function on a polar...
  15. S

    Finding Local Max/Min of Complex Function

    Homework Statement How can i determine whether a complex function has any local maximum or minimum? Homework Equations let's consider the case f(z)=z* (conjugate of z) z=x+iy The Attempt at a Solution f(z)=z*=x-iy how do i see if it has local max or min? Thank you
  16. M

    Maximum/mimimum of a complex function

    Hi, I have the following problem given a function f(k) defined on the reals and a complex constant z0, what is the maximum of the following function? z_0f(k) The maximum of the module is clearly the value k such that z_0f'(k)=0 right? because when you take the module, the squares...
  17. C

    Prove limit of complex function 2

    Homework Statement Prove using limit definition $\lim_{z \to z_0} (z^2 + c) = z_0^2 + c$. Homework Equations The Attempt at a Solution For every $\varepsilon$ there should be a $\delta$ such that \begin{align*} \text{if and only if } 0 < |z - z_0| < \delta \text{ then }...
  18. C

    Prove limit of complex function

    Homework Statement Prove that lim_{z -> 1 - i} [x + i(2x+y)] = 1 + i where z = x + iy Homework Equations Prove using definition of complex limit. The Attempt at a Solution Start from |x + i(2x + y) - (1 + i)| < epsilon, need to transform LHS to an expression that includes |(x...
  19. R

    Infinite Limit Complex Function

    Homework Statement Find the limit of a complex number involving infinity. Homework Equations Lim z->infiity of (8z^3+5z+2) The Attempt at a Solution I tried to split the problem into real and imaginary parts using z = x+iy, but I don't know how to deal with the z->infinity. Thanks!
  20. L

    Is a complex function complex diffferentiable

    Is a complex function complex diffferentiable if AND only if they are analytic? or are there counterexamples?(analytic functions which are not holomorphic) Why do you need to sometimes analyticially continue an analytic function?
  21. G

    Range of z+1/z (complex function)

    Homework Statement Give the range of the function f(z)=z+1/z z \in \mathbf{C} , Im(z)>0 Homework Equations The Attempt at a Solution tried to use polar form to either separate real and imaginary part or get r*exp(I*phi) both failed to yield suitable closed form expressions. Also tried to...
  22. G

    Finding invertible complex function

    Hi there, This is my first time posting on this site. I'm doing Calculus 2 and am stuck on finding whether or not the following functions are invertible in the given intervals and explaining why. (a) sechx on [0,infinity) --> I solved (a) but (b) and (c) is where I'm stuck. (b)...
  23. P

    Mathematica Mathematica:How can I acclaim a variable is real in a complex function

    Hi,I'm a beginner. here's the example: f[x_] := E^(I*x) Conjugate[f[x]]*f[x] I'd like get 1,but it give me a complex function coz it regard x as complex. please help me:how to claim x is real? by the way,is that necessary to define a letter as a constant.no matter yes or not,how to...
  24. Y

    Deforming Contour of complex function

    Hi, First, thank you for reading this. I've got a complex function, F(z), which is assumed to be analytic, and I know it's values along a contour in the complex plane. Say, for simplicity, that contour is known parametrically as x(t) = cos(t), y(t) = sin(t), 0 < t< pi, thus I know F(x(t) +...
  25. W

    Really complex function to integrate

    Hello, I have a really complex function to integrate (not homework), and I was wondering if there is any software application that can handle it. I have tried MathCAD and Maple, but both can't perform it. I don't think I can do it by hand, with all the integration by parts and expansions...
  26. E

    Expressing complex function in standard rectangular form

    I'm given a complex function in the exponential form: 2.5j e^(-j40*pi) Transforming this into the standard Cartesian form is pretty straight forward, but the extra j multiplying the 2.5 is kind of throwing me off. I don't know if I did it right, but this is what I did: 2.5j...
  27. G

    [Q]How can i draw real graph from complex function?

    For instance, I have a simple eigen function such as \varphi = Ae^{ik_{1}x} + Be^{-ik_{1}x} This is complex form which means we can't draw this function on real coordinate. How can i draw this function? just By taking out real term of complex function? Second question is what...
  28. B

    What is the limit of the function as z approaches 0?

    Homework Statement Used the definition of a limit to prove that as z=>0 lim (z bar)^2/(z)=0 Homework Equations abs(f(z)-w(0)) < eplison whenever abs(z-z(0)) < lower case delta The Attempt at a Solution let z=x+iy and z bar = x-iy z=(x,y) Since limit of function is approaches origin...
  29. M

    Solving Complex Function: Find Singularity of sin(sqrtZ)/Sqrt(Z)

    Homework Statement Locate & name the singularity of the function sin(sqrtZ)/Sqrt(Z) ? Homework Equations The Attempt at a Solution At z= 0 i gives 0/0 form so should i apply L hospital's rule & then proceed ?
  30. M

    Proving Complex Function Well Defined

    What does it mean to prove a complex function is well defined?
  31. J

    Simple complex function question

    Find f(z)= u(x,y) + iv(x,y), given U = x^2 - 2xy - y^2 \\ and check for analyticity. We have to find v(x,y) as follows: u_x = v_y and u_y = -v_x Cauchy-Riemann equations u_x = 2x - 2y and u_y = -(2x+2y) \\. Thereforev_y = 2x - 2y ...(i) and v_x = 2x + 2y \\ ...(ii), integrating (i)...
  32. M

    Analytic Extension on a Complex Function

    Homework Statement Prove there cannot be an analytic extension containing the unit disk of: f(x) = Series on n from 1 to infinity: x^(n!) Homework Equations Unique Extension theorem, no real explicit equations I can think of. The Attempt at a Solution So far I've proved the...
  33. B

    Proving Constant Function from Cauchy-Riemann Equations

    Problem: Suppose \Omega \in \mathbb{C} is open and connected, f is differentiable on \Omega , and f(z) \in \mathbb{R} , \ \forall z \in \Omega . Prove that f(z) is constant. Is this just a matter of solving the Cauchy-Riemann equations? If so, I think the proof is relatively...
  34. A

    Calculating Residue of a Complex Function: A GRE Problem

    In the complex plane, let C be the circle |z| = 2 with positive (counterclockwise) orientation. Show that: \int _C \frac{dz}{(z-1)(z+3)^2} = \frac{\pi i}{8} This isn't homework, it was a problem in one of the practice GREs. It looks like a straightforward application of the residue...
  35. U

    Complex Function: Real & Imaginary Parts, Square, Reciprocal & Absolute Value

    I am to find the imaginary part, real part, square, reciprocal, and absolut value of the complex function: y(x,t)=ie^{i(kx-\omega t)} y(x,t)=i\left( cos(kx- \omega t)+ i sin(kx- \omega t) \right) y(x,t)=icos(kx- \omega t)-sin(kx- \omega t) the imaginary part is cos(kx- \omega t) the...
  36. Q

    Lorentz transform as a complex function

    I tried to represent the Lorentz transform which converts a pair of space-time co-ordinates (ct,x) to (ct',x') as a function of a complex variable i.e ct' + ix' = f(ct + ix) Unfortunately the rules of complex algebra do not permit this because the complex product is defined as (a +...
  37. H

    What values of z in the complex plane make the series absolutely convergent?

    Hi, In this Problem i am finding Problem to calculate the set of z: Pls help Determine all z \subset C for which the following series is absolutely convergent: \sum (1/n!)(1/z)^n Thx
  38. JasonRox

    Maple Plot a Complex Function in maple

    How do I plot a Complex Function like... e^ix? I tried things like... complexplot(x,y); ...but that gets me knowhere. I searched on the program, but it never mentions anything about a Complex Plane of Argand Plane. I would really like to see this graph, which probably looks...
  39. L

    Seeks Assistance with n-derivative of Complex Function

    I have a problem... Can you help me ?... :confused: I need to calculate the n-derivative of a function of z (complex number)... n derivative of complex function of a complex variable Thanks, fellows ! Looker
  40. K

    Finding Optimal Parameters for Complex Function f(x)

    hi, I face the following problem. I need to find the best values of the parameters a,b,c of the complex function f(x)=a+\frac{b-a}{1+j x c} of the real variable x where (j^2=-1) such that f(2 \pi 10^6)=2.33-j 1.165 10^{-3} and f(2 \pi 10^{10})=2.347-j 3.7552 10^{-3}. It seems to be...
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