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. H

    I Complex function, principal value notation

    When a variable in ##[\text { } ]## means its principal value, ##(-\pi,\pi]##, which is correct: ##Log(z^2)=log([z]^2)## or ##Log(z^2)=log([z^2])## (both, neither)?
  2. H

    Branch points of a complex function

    My answer: one branch point is ##1## of the order 1, another is ##i## of the order 2. My question is, how can I be sure that these are the only branch points?
  3. patric44

    Is f(z) =(1+z)/(1-z) a real function?

    hi guys i found this problem in a set of lecture notes I have in complex analysis, is the following function real: $$ f(z)=\frac{1+z}{1-z}\;\;, z=x+iy $$ simple enough we get $$ f=\frac{1+x+iy}{1-x-iy}= $$ after multiplying by the complex conjugate of the denominator and simplification $$...
  4. L

    Can Cauchy's Residue Theorem be Used for Functions with Branch Cuts?

    First of all I am not sure which type of singularity is ##z=0##? \ln\frac{\sqrt{z^2+1}}{z}=\ln (1+\frac{1}{z^2})^{\frac{1}{2}}=\frac{1}{2}\ln (1+\frac{1}{z^2})=\frac{1}{2}\sum^{\infty}_{n=0}(-1)^{n}\frac{(\frac{1}{z^2})^{n+1}}{n+1} It looks like that ##Res[f(z),z=0]=0##
  5. R

    How to find the residue of a complex function

    Hi, I'm trying to find the residue of $$f(z) = \frac{z^2}{(z^2 + a^2)^2}$$ Since I have 2 singularities which are double poles. I'm using this formula $$Res f(± ia) = \lim_{z\to\ \pm ia}(\frac{1}{(2-1)!} \frac{d}{dz}(\frac{(z \pm a)^2 z^2}{(z^2 + a^2)^2}) )$$ then, $$\lim_{z\to\ \pm ia}...
  6. tixi

    Finding analyticity of a complex function involving ln(iz)

    Hey everyone! I got stuck with one of my homework questions. I don't 100% understand the question, let alone how I should get started with the problem. The picture shows the whole problem, but I think I managed doing the a and b parts, just got stuck with c. How do I find the largest region in...
  7. R

    Cauchy Riemann complex function real and imaginary parts

    Hi, I have to find the real and imaginary parts and then using Cauchy Riemann calculate ##\frac{df}{dz}## First, ##\frac{df}{dz} = \frac{1}{(1+z)^2}## Then, ##f(z)= \frac{1}{1+z} = \frac{1}{1+ x +iy} => \frac{1+x}{(1+x)^2 +y^2} - \frac{-iy}{(1+x^2) + y^2}## thus, ##\frac{df}{dz} =...
  8. C

    I Expansion of a complex function around branch point

    I’m coming at this question with a physics application in mind so apologies if my language is a bit sloppy in places but I think the answer to my question is grounded in math so I’ll post it here. Say I have a function F(z) defined in the complex z plane which has branch points at z=0 and z =...
  9. F

    Show that the real part of a certain complex function is harmonic

    Hello, I have to prove that the complex valued function $$f(z) = Re\big(\frac{\cos z}{\exp{z}}\big) $$ is harmonic on the whole complex plane. This exercice immediately follows a chapter on the extension of the usual functions (trigonometric and the exponential) to the complex plane, so I tend...
  10. S

    I Derivative of a complex function along different directions

    Below are plots of the function ##e^{0.25(x-3)^{-2}} - 0.87 e^{(x-3.5)^{-2}}## The first plot is for real values. It has a minimum at the red dot. The second plot has in its argument the same real part as the red dot, but has the imaginary part changing from -0.3 to 0.3. It shows the resulting...
  11. S

    Mathematica Derivative of the Real Part of a Complex Function (Mathematica)

    When I type in this: D [ Re[ Exp[u + 10*I] ], u ] /. u->0.5 I get this output: Of course, I could just put the Re outside and the D inside, but it would be nice to know what is wrong with the above. What's with the Re' in the output?
  12. D

    Is this complex function analytic?

    ## u_x = 3x^2 -3y^2 ## and ## v_y = -3y^2-3x^2 ## ## u_y = -6xy## and ## v_x = -6xy## To be analytic a function must satisfy ##u_x = v_y## and ##u_y = -v_x## Both these conditions are met by x=0 and y taking any value so I think the functions is analytic anywhere on the line x=0 However...
  13. Robin04

    Calculating the residue of a complex function

    The singularities occur at ##z = \pm i\lambda##. As ##\frac{d}{dz}(z^2+\lambda^2)^2|_{z=\pm i\lambda}=0##, these singularities aren't first order and the residues cannot be calculated with differentiating the denominator and evaluating it at the singularities. What is the general method to...
  14. amjad-sh

    Using Simpson's method to integrate a complex function

  15. binbagsss

    Complex function open set, sequence, identically zero, proof

    Homework Statement Hi I am looking at this proof that , if on an open connected set, U,there exists a convergent sequence of on this open set, and f(z_n) is zero for any such n, for a holomorphic function, then f(z) is identically zero everywhere. ##f: u \to C##Please see attachment...
  16. maistral

    I  Evaluation of a certain complex function

    Hi. I would like to ask regarding this function that keeps on cropping up on my study (see picture below). What I did is simply substitute values for A and b and I noticed that it ALWAYS results to a real number. If possible, I would like to obtain the "non imaginary" function that is...
  17. B

    Nyquist Plot vs. Complex Function Plot

    This is not a homework problem, I just am confused a little about the differences between a Nyquist plot and the plot of a complex function. I believe they are the same given the domain of the plot of a complex function is for all real numbers equal to or greater than zero. However, I am having...
  18. D

    MHB Prove the Following is True About the Complex Function f(z) = e^1/z

    Consider the function $f(z) = e^{1/z}$, Show that for any complex number ${w}_{0} \ne 0$ and any δ > 0, there exists ${z}_{0} ∈ C$ such that $ 0 < |{z}_{0}| < δ$ and $f({z}_{0}) = {w}_{0}$ I really don't know where to begin on this.
  19. J

    Residue at poles of complex function

    Homework Statement Homework Equations First find poles and then use residue theorem. The Attempt at a Solution Book answer is A. But there's no way I'm getting A. The 81 in numerator doesn't cancel off.
  20. Drakkith

    Derivative of a Complex Function

    Homework Statement Find the derivative of ##f(z)=\frac{1.5z+3i}{7.5iz-15}## Homework EquationsThe Attempt at a Solution I had no difficulty using the standard derivative formulas to find the derivative of this function, but the actual result, that the derivative is zero, is confusing. For real...
  21. J

    Cauchy Integral of Complex Function

    Homework Statement Homework Equations Using Cauchy Integration Formula If function is analytic throughout the contour, then integraton = 0. If function is not analytic at point 'a' inside contour, then integration is 2*3.14*i* fn(a) divide by n! f(a) is numerator. The Attempt at a Solution...
  22. T

    How do you always put a complex function into polar form?

    Homework Statement It's not a homework problem itself, but rather a general method that I imagine is similar to homework. For a given elementary complex function in the form of the product, sum or quotient of polynomials, there are conventional methods for converting them to polar form. The...
  23. F

    Where is f(z) = e-xe-iy differentiable and holomorphic?

    Homework Statement Suppose z = x + iy. Where are the following functions differentiable? Where are they holomorphic? Which are entire? the function is f(z) = e-xe-iy Homework Equations ∂u/∂x = ∂v/∂y ∂u/∂y = -∂v/∂x The Attempt at a Solution f(z) = e-xe-iy I convert it to polar form: f(z) =...
  24. Poetria

    Complex functions with a real variable (graphs)

    Homework Statement How do the values of the following functions move in the complex plane when t (a positive real number) goes to positive infinity? y=t^2 y=1+i*t^2[/B] y=(2+3*i)/t The Attempt at a Solution I thought: y=t^2 - along a part of a line that does not pass through the...
  25. T

    Classifying singularities of a complex function

    Homework Statement [/B] Find and classify the isolated singularities of the following: $$ f(z) = \frac {1}{e^z - 1}$$ Homework EquationsThe Attempt at a Solution I have the solution for the positions of the singularities, which is: ## z = 2n\pi i## (for ##n = 0, \pm 1, \pm 2, ...##) and this...
  26. T

    I What is the Limit of This Complex Function as z Approaches i?

    I am trying to find the limit of ## \frac {z^2 + i}{z^4 - 1} ## as ## z ## approaches ##i##. I've broken the solution down to: ##\frac {(z + \sqrt{i})(z - \sqrt{i})}{(z+1)(z-1)(z+i)(z-i)} ## but this does not seem to get me anywhere. The solution says ## -0.5 ## but I don't quite understand how...
  27. N

    I Help evaluating complex function in form m+ni?

    Hey all, I need the complex version of the sigmoid function in standard form, that is to say $$f(\alpha) =\frac{1}{1+e^{-\alpha}} , \hspace{2mm}\alpha = a+bi , \hspace{2mm} \mathbb{C} \to \mathbb{C}$$ in the simplified form: $$f = m+ni$$ but found this challenging, for some reason i assumed...
  28. A

    B Partial derivative of the harmonic complex function

    For a harmonic function of a complex number ##z##, ##F(z)=\frac{1}{z}##, which can be put as ##F(z)=f(z)+g(\bar{z})##and satisfies ##\partial_xg=i\partial_yg##. But this function can also be put as ##F(z)=\frac{\bar{z}}{x^2+y^2}## which does not satisfy that derivative equation! Sorry, I...
  29. M

    I Normal vector on complex function

    Hi, I'm not sure about the the normal vector N on a complex function z(x,t) = A e^{i(\omega t + \alpha x)} My approach is that (\overline{z} beeing the conjugate of z): \Re{(\mathbf{N})} = \frac{1}{\sqrt{\frac{1}{4}(\partial x + \overline{\partial x} )^2 + \frac{1}{4}(\partial z +...
  30. M

    Finding Residue of Complex Function at Infinity

    Hello everyone, I have a problem with finding a residue of a function: f(z)={\frac{z^3*exp(1/z)}{(1+z)}} in infinity. I tried to present it in Laurent series: \frac{z^3}{1+z} sum_{n=0}^\infty\frac{1}{n!z^n} I know that residue will be equal to coefficient a_{-1}, but i don't know how to find it.
  31. M

    Singularities of a complex function

    Homework Statement [/B] Find and classify all singularities for (e-z) / [(z3) ((z2) + 1)] Homework EquationsThe Attempt at a Solution [/B] This is my first attempt at these questions and have only been given very basic examples, but here's my best go: I see we have singularities at 0 and i...
  32. J

    How can the number of zeros of a complex function in a given domain be proven?

    Homework Statement Let ##D={z : |z| <1}##. How many zeros (counted according to multiplicty) does the function ##f(z)=2z^4-2z^3+2z^2-2z+9## have in ##D##? Prove that you answer is correct. Homework Equations 3. The Attempt at a Solution [/B] The function has no zeros in ##D##, which can be...
  33. W

    Entire Functions and Lacunary Values.

    #Hi All, Let ## f: \mathbb C \rightarrow \mathbb C ## be entire, i.e., analytic in the whole Complex plane. By one of Picard's theorems, ##f ## must be onto , except possibly for one value, called the lacunary value. Question: say ##0## is the lacunary value of ##f ##. Must ## f ## be of the...
  34. ion santra

    What is the Nature of Singularity in the Function f(x)=exp(-1/z)?

    what is the nature of singularity of the function f(x)=exp(-1/z) where z is a complex number? now i arrive at two different results by progressing in two different ways. 1) if we expand the series f(z)=1-1/z+1/2!(z^2)-... then i can say that z=0 is an essential singularity. 2) now again if i...
  35. M

    Use Residue Theorems or Laurent Series to evaluate integral

    Homework Statement Evaluate the integral using any method: ∫C (z10) / (z - (1/2))(z10 + 2), where C : |z| = 1 Homework Equations ∫C f(z) dz = 2πi*(Σki=1 Resp_i f(z) The Attempt at a Solution Rewrote the function as (1/(z-(1/2)))*(1/(1+(2/z^10))). Not sure if Laurent series expansion is the...
  36. C

    The Unique Limit of a Complex Function

    Homework Statement I'm struggling with the proof that the limit of a complex function is unique. I'm struggling to see how |L-f(z*)| + |f(z*) - l'| < ε + ε is obtained. Homework Equations 0 < |z-z0| < δ implies |f(z) - L| < ε, where L is the limit of f(z) as z→z0 .The Attempt at a Solution...
  37. M

    MATLAB Optimizing Plotting for Complex Functions with Large Numbers

    Hi PF! I am trying to run the following plot: k = .001; figure; hold on [X,Y]=meshgrid(-4:0.01:4); a = 5.56*10^14; b = .15/(2*.143*10^(-6)); for n = 1:8 k = k*2^(n-1); Z = a./(X.^2+Y.^2).*exp(b.*(X-sqrt(X.^2+Y.^2)))-k; contour3(X,Y,Z) end which works great if a = b = 1. But now...
  38. W

    MATLAB Matlab summation of a complex function

    Hi, I need to plot the last function of this: But I don't know how to generate the sum. I know the for loop is totally wrong, but I can't go any further. This is what I have: Can someone fix the summation loop part for me? Thanks in advance
  39. blue_leaf77

    Steepest descent vs. stationary phase method

    Up to this point I have got a grasp of some basics of "steepest descent method" to evaluate the integral of a complex exponential function ##f(z) = \exp(A(x,y))\exp(iB(x,y))##. Using this method the original integration path is modified in such a way that it passes through its saddle points...
  40. S

    Find the maximum value of this complex function

    Homework Statement Find the maximum value of f(z) = exp(z) over | z - (1 + i) | ≤ 1 Homework Equations |f(z)| yields the maximum value The Attempt at a Solution f(z) = exp(x) ( cosy + i siny) Unfortunately that's all I've got. I've seen examples with polynomials, but not with trigonometric...
  41. B

    MHB Complex function that satisfies Cauchy Riemann equations

    Hi, I am currently teaching myself complex analysis (using Stein and Shakarchi) and wondered if someone can guide me with this: Find all the complex numbers z∈ C such that f(z)=z cos (z ̅). [z ̅ is z-bar, the complex conjugate). Thanks!
  42. B

    Integrating a Complex Function Over a Contour

    Homework Statement ##z(t) = t + it^2## and ##f(z) = z^2 = (x^2 - y^2) + 2iyx## Homework EquationsThe Attempt at a Solution Because ##f(z)## is analytic everywhere in the plane, the integral of ##f(z)## between the points ##z(1) = (1,1)## and ##z(3) = (3,9)## is independent of the contour (the...
  43. F

    Problem integrating complex function

    Homework Statement Hello, I have been tasked with the next problem, I have to prove that the next two integrals are complex numbers; but I have no idea of how to attack this problem. Homework Equations ∫dx f*(x) x (-ih) (∂/∂x) f(x) integrating between -∞ and ∞ ∫dx f*(x) (-ih) (∂/∂x) (x f(x))...
  44. T

    Singularities of a Complex Function

    Homework Statement What are the region of validity of the following? 1/[z2(z3+2)] = 1/z3 - 1/(6z) +4/z10 Homework EquationsThe Attempt at a Solution Knowing that this is the expansion around z=0, I am trying to find the singularities of the complex function. Which is when z2(z3+2) = 0 I...
  45. B

    Directional Derivative of Complex Function

    Homework Statement We are given that ##f(z) = u(x,y) + iv(x,y)## and that the function is differentiable at the point ##z_0 = x_0 + iy_0##. We are asked to determine the directional derivative of ##f## 1. along the line ##x=x_0##, and 2. along the line ##y=y_0##. in terms of ##u## and...
  46. B

    Verifying Whether A Complex Function Is Differentiable

    The problem is to determine whether the function ##f(z) = \left\{\begin{array}{l} \frac{\overline{z}^2}{z}~~~if~~~z \ne 0 \\ 0 ~~~if~~~z=0 \end{array}\right.## is differentiable at the point ##z=0##. My two initial thoughts were to show that the function was not continuous at the point...
  47. B

    Finding the Limit of a Complex Function

    Hello everyone, How do I find the limit of a complex function from the definition of a limit? For instance, consider the limit ##lim_{z \rightarrow -3} (5z+4i)##. Would I simply conjecture that ##5z + 4i## approaches ##5(-3) + 4i## as ##z \rightarrow -3##; and then use the definition of a...
  48. PeteyCoco

    Zeros of complex function in SciPy

    I've been told that the method scipy.optimize.Newton() will solve complex functions so long as the first derivative is provided. I can't make it work. The documentation for Newton() mentions nothing of complex functions. Could someone show me how one would find the roots of a function like f(z)...
  49. PeteyCoco

    Finding the zeros of a complex function in PyLab

    I have this characteristic equation for the wave number eigenvalues k_n of a homogeneous infinite cylinder of radius R: D_{m} = (k_n R) = 0, where D_m (z) = n_r J'_m(n_r z)H_m(z) - J_m(n_r z)H'_m(z) and n_r is the refractive index of the cylinder, the bessel and hankel functions are...
  50. S

    Integrate complex function over unit circle

    Homework Statement Calculate ##\int _Kz^2exp(\frac{2}{z})dz## where ##K## is unit circle.Homework Equations The Attempt at a Solution Hmmm, I am having some troubles here. Here is how I tried: In general ##\int _\gamma f(z)dz=2\pi i\sum_{k=1}^{n}I(\gamma,a_k)Res(f,a_k)## where in my case...
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