# Tagged Questions

Questions about using complex numbers in Mathematica. This includes basic arithmetic, functions of complex numbers, plotting complex functions, and dealing with branch cuts.

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### Plotting Complex Quantity Functions

Trying to plot with complex quantities seems not to work properly in what I want to accomplish. I would like to know if there is a general rule/way of plotting when you have complex counterparts in ...
8k views

### Finding real roots of negative numbers (for example, $\sqrt[3]{-8}$)

Say I want to quickly calculate $\sqrt[3]{-8}$, to which the most obvious solution is $-2$. When I input $\sqrt[3]{-8}$ or Power[-8, 3^-1], Mathematica gives the ...
10k views

### How to calculate contour integrals with Mathematica?

How to calculate the integral of $\frac{1}{\sqrt{4 z^2 + 4 z + 3}}$ over the unit circle counterclockwise for each branch of the integrand?
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### How to specify assumptions before evaluation?

If I request mathematica evaluate an integral for me, I'll often get a more general ConditionalExpression than I want. Example : ...
7k views

### How can I generate this “domain coloring” plot?

I found this plot on Wikipedia: Domain coloring of $\sin(z)$ over $(-\pi,\pi)$ on $x$ and $y$ axes. Brightness indicates absolute magnitude, saturation represents imaginary and real magnitude. ...
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### How does Mathematica understand branchcuts of the complex logarithm?

Say I have the function $f(x) = x \tanh(\pi x) \log (x^2 +a^2)$ where $a$ is some positive real number. Then it seems to be me that Mathematica when given such a ...
5k views

### Plotting complex Sine

I've got another plotting problem. I want to plot Sin[z] where z is complex. So, I've tried the following: ...
6k views

### Plotting complex numbers as an Argand Diagram

I have the function: $F(\omega) = \frac{5\; - \;i\;\omega}{5^2\; +\; \omega^2}$ When $\omega$ has the values : $\{ -7, -2,\; 0,\; 2,\; 7\}$ How would I plot the Argand diagram in Mathematica? Or ...
3k views

### How can I convert a complex number into an exponent form

When I have an expression such as (1/4 + I/4) ((1 - 2 I) x + Sqrt[3] y) it is hard to get an intuition of the number. So I want to convert it to the complex ...
290 views

### When does the real part of Zeta vanish on the critical line?

This seems to be a quite a simple problem but I cannot make it work. I am trying to find all values within a given range for which the real part of the Zeta ...
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I've been playing around with a visualization technique for complex functions where one views the function $f: \mathbb{C} \rightarrow \mathbb{C}$ as the vector field $f: \mathbb{R^2} \rightarrow \... 4answers 9k views ### Factoring polynomials to factors involving complex coefficients I've run into some problems using Factor on polynomials with complex coefficient factors. Reading the documentation it looks like it only factors over the ... 2answers 736 views ### How to eliminate the zero real part of a purely imaginary number? In Mathematica 9, a purely imaginary number, e.g. 0.9 I, will display as 0. + 0.9i in the output form. How can I eliminate the ... 2answers 3k views ### Paths integrals in the complex plane I can't find how to calculate path integrals of complex functions in the complex plane. For example: $$\oint_{\mid z \mid =2}\frac{1-e^z+z}{z^3 (z-1)^2}dz$$ 3answers 4k views ### Derivative of real functions including Re and Im When deriving functions using Re, Im or Arg (and probably some other functions as well), ... 1answer 2k views ### Complex line integral Can someone recommend an online article or introductory tutorial that will show me how to do real and complex line integrals using Mathematica? 3answers 205 views ### Complex result for Real vectors in VectorAngle I was expecting a real angle using VectorAngle when passing real valued vectors, but I obtained a complex angle: ... 3answers 238 views ### Wrapper for inexact numeric complex numbers that maintains polar form Related question: How can I convert a complex number a+b I to the exponent form A Exp(I phi)? Mathematica insists on displaying complex number in form a+I b when <... 1answer 1k views ### Finding residues of multi-dimensional complex functions Say I have a function$f$of$n$complex variables,$\{ z_i \}_{i=1}^{i=Nc}$. And then I want to contour integrate the expression such that for each$z_i$its an integration on an unit circle about ... 3answers 5k views ### Is there a simple way to plot complex numbers satisfying a given criteria I think this should be straightforward, but I cannot seem to find a good source on how to do it after searching around, so I'm trying to sketch sets of complex numbers that meet a given for criteria. ... 1answer 1k views ### Symbolic integration in the complex plane Context While answering this question, I defined (symbolic and numerical) path integrations as follows ... 2answers 804 views ### Does FindFit support complex numbers or doesn't it? Inspired by this previous question: Findfit doesn't give the good fit; Changing the starting values will not change the results. Consider the following complex-valued dataset. ... 1answer 506 views ### Numerical inverse Laplace-Hankel transform When trying to reproduce the result of this paper about numerical solution of Lamb's problem, I encountered the following double integral (to be more precise, the 0-order inverse Hankel-Laplace ... 2answers 658 views ### Draw the image of a complex region I'm working on a complex question that asks that I determine a function that maps the complement of the region$D=\{z:|z+1|\le 1\}\cup\{z: |z-1|\le 1\}$onto the upper half plane. That is,$f$must ... 1answer 3k views ### Is it possible to set a variable as a positive one in the whole notebook? I'm having issues during integration due to the fact that Mathematica doesn't know if an undefined variable is positive or not (it gives me complexes which bothers me in the end). For example I do ... 3answers 1k views ### Complex conjugate How do I complex conjugate a vector? ... 1answer 789 views ### Does the Im function work with symbolic arguments? Does the Im function work with symbolic arguments? ... 2answers 477 views ### Eigensystem, Eigenvalue doesn't output nonreal eigenvalues Basically I have a matrix and when I used either Eigenvalue or Eigensystem, it doesn't output nonreal eigenvalues, instead it ... 1answer 1k views ### why there is a small imaginary part [closed] I encountered a problem. I have a eigenvector eigvsI[1] ... 1answer 3k views ### How to visualize Riemann surfaces? In WolframAlpha we can easily visualize Riemann surfaces of arbitrary functions, can we plot the Riemann surface of an arbitrary function using Mathematica and ... 1answer 3k views ### Bifurcation diagrams for multiple equation systems I am interested in constructing a bifurcation diagram for some of my parameters (especially for Î² and Î³) in the dynamical system given in the code below. I want to see how parameter changes affect the ... 2answers 9k views ### How to tell Mathematica that certain variables are real/imaginary, integer-valued, etc I'm trying to expedite some quantum mechanical calculations by running them through Mathematica. Expectation values and stuff like that. When I say, for example, ... 1answer 5k views ### Complex number operations: telling Mathematica variables are real I want to do Conjugate[a + b*I], but when I do that, the solution is Conjugate[a] - I*Conjugate[b]; when for me, a and b are ... 1answer 661 views ### How do I put an image on the complex plane? I watched this video and became interested in transforming an image. But I have no good idea on how to embed an image in the complex plane using Mathematica. I have a method that seems to work, but ... 1answer 374 views ### Factoring a two variable polynomial in a special way Let $$f=x^9-x^6+4x^5y+2x^3y^2-y^4$$ I would like to factorize$f$into form: $$(y-F_1(x))\cdots(y-F_k(x))$$ over complex numbers. How can I do it with Mathematica? 3answers 544 views ### Conjugating symbolic expression I am trying to conjugate a symbolic expression, and I have explicitly stated the real terms. However, I simply can't get it to work: ... 1answer 510 views ### Working with derivative of conjugate of a complex number I have a complex function, lets say$g(x)$. I want to take its and its conjugate's derivative. I need the solution of derivative which must be symbolically and computationally efficient. Lets take an ... 1answer 135 views ### Residue failed to reproduce this residue in a paper This problem is a more clearly state of my original problem which is off topic, so I reconsider my problem carefully based on previous discussions, and gives the following valid code to reproduce my ... 1answer 214 views ### How to reduce expressions with complex coefficients in the form of a+0.*I I'm trying to get an eigenvalue equation in Mathematica, and the result is an expression with coefficients of the form a + 0. I. For example, Is there any clever ... 2answers 331 views ### presenting a real number as real instead of imaginary I have an equation which results in an answer of the form$\frac{i a}{\sqrt{c-d}}$is there any way to get Mathematica to present it in it's real form? like$\frac{a}{\sqrt{d-c}}$I know that I ... 2answers 822 views ### What can I do to eliminate the error FindFit::nrjnum:? I am testing the "Power Law with finite-time singularity" hypothesis for world population growth for a project. The data I'm using (same behaviour should also be exhibited by the stock market, thats ... 2answers 1k views ### Why does this integral have a complex component? I wanted to find the probability of my normally-distributed random variable being at least 15, so I set up this integral: ... 1answer 243 views ### Derivative of conjugate multivariate function I have a problem with Mathematica, taking the derivative of the conjugate of some function. I know that a similar question has been posed before here, but the solution did not work for multivariate ... 0answers 129 views ### Evaluating Real and Imaginary parts in a dispersion equation via ContourPlot [closed] I want to reproduce a dispersion curve from P. Zou's thesis found here: http://www.ipd.anl.gov/anlpubs/2001/06/39620.pdf on Pg. 31, Eq. 2.13. The problem is, the code below only gives values for the ... 1answer 792 views ### Dual complex integral over implicit path using contour plot Context I am interested in doing double contour integral over paths which are defined implicitly. For the sake of debugging, let's assume its $$\oint_{\cal C}\oint_{\cal C} \frac{1}{u\, x} d u d x$$ ... 6answers 2k views ### Image of first quadrant under$f(z)=(z+i)/(z-i)$I'm able to plot the region where Im[z] > 0 and Re[z] > 0: ... 2answers 2k views ### Plot3D with coloring based on phase value of f I'm reading a sample chapter of Visual Complex Functions by E. Wegert. Near the bottom of this page, there is a link "Download Sample pages 1 (pdf, 2.4 MB)" which allows you to read Chapter 2 of the ... 3answers 7k views ### How can I plot the complex graph of$x^x$in Mathematica? For example, this Wolfram Alpha query shows this graph: But it does not show the code for plotting it in Mathematica. Plot[x^x, {x, -1, 1}] only plots the real ... 1answer 1k views ### Stereographic Projection Say I want to represent points of the complex plane in the sphere$\Bbb S^2$using stereographic projection. That is, the Riemann sphere: Specifically, it would be nice to be able to: Given the ... 3answers 344 views ### Riemann-Spherical Projection of Network onto the$\mathbb{R}^2\$-Plane

I'm trying to find a way to plot the graph ...