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.

132 questions with no upvoted or accepted answers
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460 views

Expansion for Modified Bessel Function Around Infinity

I'm somewhat new to Mathematica, and I don't understand why I'm getting inconsistent series expansions for the modified Bessel Function of first kind near $x=\infty$. First problem: I get different ...
6
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0answers
68 views

Where is the mistake in computing the particular eigenvector of the following DFT Matrix?

I have the following matrix (the DFT Matrix for N = 3) $$W = \frac{1}{\sqrt{3}}\begin{pmatrix} 1 & 1 & 1 \\ 1 & e^{-\frac{i 2 \pi}{3} } & e^{\frac{i 2 \pi}{3} } \\ 1 & e^{\frac{...
5
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164 views

Find regions in which the roots of a third degree polynomial are real

I have to find the roots of a third degree polynomial in $\phi$ that depends from 3 parameters, namely $t,s,w\in \mathbb R$. In order to do that I've used the command ...
4
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0answers
67 views

Prime factorization over the Eisenstein integers $\mathbb{Z}[\zeta]$

I am trying to write a function f[a_,b_] which takes in two integers $a,b$ and returns the unique factorization of $a+be^{2\pi i/3}$ into the primes belonging to $\...
4
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0answers
69 views

Is `InverseMellinTransform` unaware of second Barnes lemma?

Consider evaluating ...
4
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0answers
125 views

Why isn't Mathematica listening to my assumptions?

I'm trying to simplify the expression ωplus := 1/2 c (I f V + Sqrt[4 k^2 - f (f + 8 I k) V^2]) so that I can get it in the form $a+i b$. I am using the global ...
4
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0answers
66 views

Why does FreeQ seem not to work with complex numbers?

I guess I do not understand how Mathematica evaluates this simple expression inside FreeQ ...
3
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3answers
576 views

Plot a specific branch of a multi-valued complex function

Let's say I want to plot a branch of the following function $f(z) = \sqrt{z(z-1)}$. Let $z = r_1 e^{i\theta_1}$ and $z-1 = r_2 e^{i\theta_2}$. How can I tell Mathematica to plot the branch for which $...
3
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0answers
113 views

Weird behavior of PolyLog

I have asked Mathematica (11.1.0.0) to calculate Sum[Log[n]/n (-9/10)^n, {n, 2, ∞}] This is the sum $\sum_{n=2}^{\infty}\limits\frac{\log n}{n}\left(-\frac{9}{10}...
3
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0answers
110 views

Solving a differential equation with a complex (independent) varible

I'm trying to plug in complex values in the numerical solutions of ODEs without success. For instance y[I] /. NDSolve[{y'[x] == 0, y[0] == 3}, y, {x, -10, 10}] ...
3
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0answers
825 views

Discrepancy in the series expansion of $\log(z/(z-1))$ in Mathematica 5

When I calculate the series expansion of $\log\frac{z}{z-1}$ around $0$ in Mathematica 5.2, I obtain: ...
3
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0answers
687 views

Taking real and imaginary parts of indexed functions and speeding up ComplexExpand

I am setting up a large system of ODEs and in order to use the IDA method (which is sig. faster for my system and thus attractive), I must split my equations into real and imaginary parts. I am ...
2
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0answers
78 views

How to convince Mathematica that $(-1)^{2/3} = e^{2\pi i /3}$

I have a list T which contains 3 + 4*E^(2*Pi*I/3). Yet MemberQ[ T , 3 + 4*(-1)^(2/3) ] ...
2
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0answers
115 views

Plotting from the celebrated Euler's formula

ContourPlot[ x^(I*y) == -1, { x, 1, 3}, {y, 1, 3}] A newbie's query..The most famous equation should ( methinks) yield a lot of points other than the real pair $(...
2
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0answers
44 views

How is a Lévy stabile process pdf normalized?

I'm interested in how to deal with α-stabile probability density functions, for example one with the Laplace image like: $ \phi(s) = s^{(\alpha-1)}*e^{-\nu s^{\alpha}} $ I can work with this ...
2
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0answers
127 views

Numerical estimates on asymptotic of given functional as $x\rightarrow \infty$

Consider the following functional : $$ I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy, s) − F(x −\mathrm iy, s)}{\mathrm e^{2πy}-1}, $$ where $ F(z, s) = \dfrac{\sin^2[π\Gamma(z)/(2z)]}{z^s} ...
2
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1answer
59 views

How to chop complex number from solution solve using a model

Hi I have a set of 4 solution that was solved using a 4th order polynomial equation. I have noticed that some of the real number solution occurred through the list of 4 set of solutions. I have ...
2
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0answers
315 views

Speed up Mathematica's Simplify and ComplexExpand functions

I am trying to simplify an expression and taking the real part only. However Mathematica use ages to run the command (if it is able to do so in the end). The problem is that the expression I am ...
2
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0answers
38 views

Specifying different thickness for each graded contours of Mandelbrot Set so that I can export it as .STL file.

I am using Mandelbrot Set function in Mathematica to get the Mandelbrot plot. I want to 3d print this model, so want to specify thickness for each contour and then convert into .STL file. Is this ...
2
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0answers
132 views

Integral on unit disk of a function with two singularities

The goal is to prove, using Mathematica, that for $\mathbb{D}$ the unit disk and $u,v \in \mathbb{D}$, $u \neq v$, $$\frac{1}{\pi} \int_{\mathbb{D}} \frac{\mathrm{d}^2z}{(z-u)\overline{(z-v)}} = \ln(1-...
2
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0answers
41 views

Linearity of the Conjugate operator

The output of Grad[Conjugate[x + y + z], {x, y, z}] is ...
2
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0answers
63 views

How MandelbrotSetBoettcher function is computed?

Boettcher function is a solution of Boettcher's functional equation, so in case of complex quadratic map: $B(f(z)) = (B(z))^2$ where: B is Boettchers function f is complex quadratic map There is a ...
2
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0answers
27 views

`FunctionDomain` working with real part

Is there any reason why Mathematica evaluates the following FunctionDomain[Re[((I + 1) x)], x] (* False *) as False, where I'...
2
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0answers
79 views

Amplitude of a transfer function

I'm looking to calculate the amplitude of a transfer function which is defined in a symbolic way : T=(1 + 2 I r λ)/(1 - r^2 + 2 I r λ) I try this, i doesn't work ...
2
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0answers
98 views

How to ComplexExpand Jacobi elliptic functions?

It seems that Mathematica cannot expand Jacobi $\text{sn}, \text{cn}, \text{dn}$ functions. For example, A&S 16.20.1-3 say that for real $y$ $$ \operatorname{sn}(i y \mid m) = i\operatorname{sc}(y ...
2
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0answers
360 views

Padé approximation of a complex function

I have an interesting problem for the Mathematica. I have some function, let's call it $G$. This is function of discrete set of equidistant points on imaginary axis, $i \omega_n$, so the typical data ...
2
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0answers
329 views

Using Arnoldi Method with Multiple Options

I have a matrix with many complex eigenvalues, but I only need a few near a particular complex number. I am only looking at the imaginary parts, so I need the few closest on the imaginary scale to my ...
2
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0answers
155 views

Constrain integration to be over real domain

I am trying to integrate the following: Integrate[(2*Cos[Sqrt[F]*z - G])/E^(D*(p + z)^2), {z, -(h/2), h/2}] I am getting solution in complex domain: ...
2
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0answers
88 views

The number of real values assumed by combinations of complex roots of a decic

Define complex conjugate as the pair of complex numbers $a+bi,a-bi$. Assume you have 5 such pairs and they are the 10 roots $x_i$ of a 10th-deg equation with integer coefficients. In general, how many ...
2
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0answers
421 views

Simple contour integral with a parameter gone wrong

Bug introduced in 7.0 and fixed in 7.0.1 I run into the following problem, I tried to evaluate a very simple integral: ...
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0answers
35 views

Integrate returning complex values

I am having trouble using mathematica to calculate some integrals Namely this : ...
1
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1answer
41 views

Is it possible to have a complex nu in ParabolicCylinderD?

I'm trying to recreate a graph from a paper, and the function that I'm plotting involves three separate parabolic cylinder functions that all have a complex Nu value. After getting it all typed out, I ...
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0answers
43 views

How do you define the domain and range of the variables for a plot?

How do I define the domain to be only reals and the range to be only complexes? Everytime I plot using ComplexPlot[], I get a C->C function so there are basically too many variables to plot and I ...
1
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0answers
41 views

Ordinary differential equation in the real domain

If I have the following equation (y - 1) ^5 == 32 I can select the real solution with Solve[(y - 1) ^5 == 32, x, Reals] Now, I ...
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0answers
42 views

What are “VariableReplacementDump” and “TranscendentalDump”?

I have run into a Mathematica object I've never seen before and Googling those reveals nothing. I'm talking about VariableReplacementDump and ...
1
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0answers
55 views

Simplification of expression with sum of absolute values

I try to simplify following expression 1/2*(Abs[a1] + Abs[b1] + Abs[c1] + Abs[d1]), where a1, b1, c1, d1 are quite long: ...
1
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0answers
44 views

Change the PolyLog branch cut in Wolfram Mathematica

The problem is to change the branch cut of PolyLog[2,z] function from the default half line z=1+x, x>0 to the new one ...
1
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0answers
39 views

Mathematica cannot evaluate the real part of a fraction?

I am encountering some confusing behaviour with Mathematica that maybe you can help me resolve: If I type: Im[1/a]~Refine~(a \[Element] Reals) I get the output: ...
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0answers
34 views

Solving ODEs in the Complex Plane for General z

I saw with the new 12.0 update, they have introduced expanded functionality for solving complex ODEs. In the given example, they solve for $y(t)$ and plot results using the new ...
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0answers
30 views

Errors in NDSolve for complex matrices

This is the problem. It is quite simple, and looks unwieldy only because I had to rewrite quaternions as complex matrices. ...
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0answers
58 views

Defining dilogarithm with branch cut on $(-\infty,1)$

If my logarithms have a branch cut on $(0,\infty)$ then the dilogs constructed from these have a branch cut on $(-\infty,1)$. Is there a safe way to implement this in Mathematica? I tried with ...
1
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0answers
33 views

How to compute Bott-Chern operator ddc

I wish I could make Mathematica compute the $dd^c=2i\partial\bar\partial$ of a function depending on several complex variables. The question involves complex or Wirtinger derivatives (https://...
1
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0answers
81 views

How do the Around and AroundReplace commands work with the complexes?

Let us consider s = Around[1, 0.2] + I*Around[2, 0.3] Around[ Complex[1., 2.], 0.36055512754639896`] What does it mean? The documentation to the Around ...
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0answers
80 views

How can I get Mathematica to recognize that $ab^* + ba^* = 2\Re(ab^*)$?

Mathematica understands that $z + z^* = 2\Re(z)$, FullSimply[z + Conjugate[z]] (* output: 2 Re[z] *) However, Mathematica does not simplify the expression ...
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0answers
49 views

Discontinuities in taking the roots of a complex function

I have the following functions e1, e2, and e3 which consists of the functions ...
1
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0answers
70 views

Efficient evaluation of real/imaginary parts of long expressions

I have the following, rather compact, but complex, expression (see below). I simply want the real part of this. Now, when I do the usual, i.e., ...
1
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0answers
38 views

Inverting a series

How do I invert the following, \[Rho]=r + b0 Sum[Pochhammer[1/2, k]/(k! ((1 - q) k - 1)), {k, 0, \[Infinity]}] + b0^(1 - q)/(2 q) r^q + O[r^(2 q - 1)] to get $r(...
1
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0answers
101 views

Is there a way to reduce size of complex number?

I would like to create huge matrices with hundreds of thousands of elements per side and I would like to use complex numbers in it for further processing by Mathematica functions. When I investigate ...
1
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0answers
334 views

How to graph this function's Riemann surface with Mathematica

I was studying Riemann surfaces, when I found this fancy plot Sadly the only specification of the figure was this miser caption Riemann surface for $$W(z)^{d}+W(z)+z^{d-1}$$ with $d=5$ where $W(...
1
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0answers
62 views

CountRoots[] won't count my roots. Can I force it to?

I have a big messy function consisting of lots of combined Bessel and Hankel functions that I want to count the roots of. Mathematica doesn't want to try though because it can't prove the function is ...