# Questions tagged [complex]

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

Say I want to quickly calculate $\sqrt{-8}$, to which the most obvious solution is $-2$. When I input $\sqrt{-8}$ or Power[-8, 3^-1], Mathematica gives the ...
5answers
40k views

### 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 ...
2answers
18k 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?
5answers
11k 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. ...
6answers
8k 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 ...
3answers
7k 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 ...
2answers
15k views

### 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 : ...
2answers
1k 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 ...
2answers
7k 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
42k views

### How to tell Mathematica that certain variables are real/imaginary, integer-valued, etc

I'm trying to expedite some quantum mechanical calculations (expectation values etc.) by running them through Mathematica. When I say, for example, ...
4answers
14k 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 ...
3answers
412 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: ...
1answer
3k views

### 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 ...
3answers
5k views

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 \... 3answers 6k 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 y) it is hard to get an intuition of the number. So I want to convert it to the complex ... 2answers 462 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 ... 3answers 7k views ### Derivative of real functions including Re and Im When deriving functions using Re, Im or Arg (and probably some other functions as well), ... 2answers 349 views ### What is the best way to define Wirtinger derivatives Wirtinger derivatives ( also called Cauchy operators) in complex analysis are widely used tools. They are defined in the case of one dimensional complex plane as follows $$\frac{\partial}{\partial z}=... 3answers 289 views ### Wrapper for inexact numeric complex numbers that maintains polar form Related question: How can I convert a complex number into an exponent form Mathematica insists on displaying complex number in form a+I b when ... 2answers 1k 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 ... 1answer 8k 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 ... 2answers 11k 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: ... 1answer 3k 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? 1answer 991 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 ... 3answers 6k 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. ... 2answers 1k views ### Why doesn't FullSimplify drop the Re function from an expression known to be real? For some reason Mathematica does not properly simplify this expression: ... 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 ... 2answers 2k 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 414 views ### Orbit followed by a particle around Schwarzschild Black Hole The following is a equation which describes various possible orbits of a particle around the Schwarzschild black hole spacetime in general relativity. I want to solve it from ... 3answers 2k views ### Complex conjugate How do I complex conjugate a vector? ... 1answer 542 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))$$over complex numbers. How can I do it with Mathematica? 2answers 715 views ### Eigensystem, Eigenvalue doesn't output nonreal eigenvalues Basically I have a matrix and when I used either Eigenvalue or Eigensystem, it doesn't output nonreal eigenvalues, instead it ... 1answer 2k views ### Symbolic integration in the complex plane Context While answering this question, I defined (symbolic and numerical) path integrations as follows ... 4answers 695 views ### Why is (-1.)^2. a complex number Why (-1.)^2. in Mathematica returns a complex number? It looks like in both C and Fortran it returns 1. Why does Mathematica behave differently than the other ... 2answers 2k views ### Does FindFit support complex numbers or doesn't it? Inspired by this previous question: Findfit doesn't give the good fit; Changing the starting values will not change the results. Consider the following complex-valued dataset. ... 1answer 1k views ### How do I put an image on the complex plane? I watched this video and became interested in transforming an image. But I have no good idea on how to embed an image in the complex plane using Mathematica. I have a method that seems to work, but ... 2answers 2k views ### How to tell Mathematica that the argument of a function is real? I want to define the following function in mathematica:$$R(\psi)=\begin{bmatrix}1&0&0\\0&\cos 2\psi & \sin 2\psi\\0&-\sin 2\psi & \cos 2\psi\end{bmatrix}\qquad,\psi\in\mathbb ... 2answers 919 views ### Visualizing a holomorphic bijection between the unit disc and a domain One can construct a holomorphic bijection between the (open) unit disc$D=\{z\in{\bf C}: |z|<1\}$, and a domain$D\setminus\overline{B_{1/2}(-1/2)}$where$B_r(z_0)$denotes the ball of radius$r$... 2answers 886 views ### Compiling the VoigtDistribution PDF According to List of compilable functions, Erf and Erfc are compilable functions. However, I want to make a compiled version ... 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 ... 1answer 132 views ### Re and Im relationships without Abs among Fourier, FourierTransform, InverseFourier, and InverseFourierTransform Is there a way to make Fourier's Re and Im components separately track the results of ... 1answer 292 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 ... 1answer 974 views ### Does the Im function work with symbolic arguments? Does the Im function work with symbolic arguments? ... 1answer 2k views ### How to make the imaginary part of a +0. I zero globally? Values like a +0. I are really annoying. Answers from How to reduce expressions with complex coefficients in the form of a+0.*I？ Is there a way to globally set when to treat a very small number as ... 1answer 3k views ### why there is a small imaginary part [closed] I encountered a problem. I have a eigenvector eigvsI ... 0answers 76 views ### Instructing DSolve not to return answers with explicit complex numbers DSolve sometimes returns solutions containing explicit complex numbers, even though the ODE does not. For instance (drawn from question 126072), ... 2answers 3k views ### Möbius transformations revealed Möbius Transformations Revealed is a short video that vividly illustrates the simplicity of Möbius transformations when viewed as rigid motions of the Riemann sphere. It was one of the winners in the ... 4answers 2k views ### Visualizing the Riemann zeta function How to visualize in a nice geometric way (e.g., like in this YT video) the Riemann zeta function,$\zeta (z)$? 3answers 2k views ### How can I recreate Trott's Riemann Surface plot in Mathematica? In reading Michael Trott's Visualization of Riemann Surfaces of Algebraic Functions, he has: ... 5answers 12k 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 ...