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.
1,308
questions
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Trouble with Finding a Supremum Constrained by a Condition
I have a function
$$
f(x,y) = \sqrt{\left(x^4 y+\frac{2 x^2 y^2}{x^2 y^2+1}+\frac{2}{x^2 y^2+1}-41 x^2 y-7 x^2\right)^2+\left(-7 x^3 y-x^3-\frac{2 x y}{x^2 y^2+1}-\frac{2 x^3 y^3}{x^2 y^2+1}+41 x\...
3
votes
0
answers
90
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Workarounds for unevaluated results of ContourIntegrate of non-analytic functions
There is a new useful command ContourIntegrate since 13.3.
I find that the command is well-done, but even the Sun has spots.
...
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21
votes
2
answers
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How can we reproduce Eugen Jahnke's drawing of a complex function?
Eugen Jahnke (1863 - 1921) studied mathematics and physics at the Humboldt University of Berlin. He became a professor at the Berlin Mining Academy in 1905.
His "Tables of Functions with ...
- 49.2k
3
votes
1
answer
243
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Absolute value of a complex number [closed]
I want to calculate |xx|^2.
Why can't Mathematica just calculate it?
What am I supposed to do here?
I just want it to actually calculate it, so that there are no imaginary numbers left.
Is that so ...
- 33
1
vote
3
answers
112
views
Plotting |z-2| < 1 under w = z/(z-1)
How can I plot the image of |z-2| < 1 under the transformation w = z/(z-1)? The expected result should look something like ...
- 517
1
vote
0
answers
36
views
How to assume a function is holomorphic?
For example, for a holomorphic function $f(z)=f(x,y)$, it's well know that $\ln |f|$ is harmonic. I want to check it with mathematica with the following code:
...
- 11
16
votes
1
answer
642
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How can we plot Man Ray's "Shakespearean Equation" w == e^(1/z)?
1. Man Ray
Man Ray (1890 - 1976) was an American artist who spent most of his career in Paris. He was a significant contributor to the Dada and Surrealist movements. Man Ray produced major works in a ...
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23
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Simplify vector calculus based on component splitting
I am interested in verifying a few Vector Calculus expressions using Mathematica.
These expressions are based on splitting a vector field into its transverse components (in the xy- plane) and ...
4
votes
2
answers
194
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Complex region plot
How do I plot the following two sets in the complex plane:
$C=\{z\in\mathbb{C}|z^4\in[0,16]\}$
and the set of solutions of equation
$Re(\frac{2z+1}{z+1})>0$
Thanks in advance.
- 377
2
votes
2
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182
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Real zeros of complex function
I must find the zeros of a complex function $\xi$ that depends upon two real variables $B$ and $k$. The function is obtained by evaluating the determinant of the identity matrix of dimension $2n$ ...
- 121
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2
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69
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How Not to Simplify Exponential of Log?
I have the expression
Power[x,I y]
Both $x,y$ are real. Now I want to write this expression in term of Exponential of Logarithm. The expression can be rewritten as
...
- 698
4
votes
1
answer
185
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Implementation of SINC filter using Integrate results in incorrect output
Bug introduced in 6.0, persisting through 13.3.1.
This is SINC filter, integrate $|H|^2$ from $-\pi/2$ to $\pi/2$.
MMA CODE, Result = 7.27956e-12
...
- 127
0
votes
0
answers
51
views
How does Mathematica select principal roots of complex numbers?
When I do
I^(1/3)
I get
Out[]:= (-1)^(1/6)
which is one of the solutions of $z^3=i$. There are two more solutions which are ...
- 4,179
0
votes
2
answers
78
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How to draw a 3D plot of a function which have two parameters with complex parametric values?
I am trying to draw the 3D figure of the function 1+e^{px+qy} with the complex parametric values p=1+i and q=2-3i. Actually, I am trying to draw a 3D figure of a function of the form f(x,y) = e^{ax+by}...
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61
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Visualizing conformal map from upper half plane to n-gon
I want to plot both the upper half plane and the image of it i.e. a polygonal region via Schwarz-Christoffel maps. One example is for a quadrilateral.
...
- 417
2
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0
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143
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Finite sum of complex exponentials giving wrong result [closed]
I was playing with the following sum:
$$ \sum_{k=0}^{2} e^{\frac{2\pi i k^2}{3}} = e^{0} + e^{\frac{2\pi i}{3}} + e^{\frac{8\pi i}{3}} \ne 0, $$
which is non-zero, at least according to Wolfram|Alpha.
...
- 133
2
votes
1
answer
98
views
Level curves of $|e^{1/z}|= s$ for $s \in \mathbb{R}$
In order to find the behavior of a complex-valued function at an essential singularity, I want to plot $|e^{1/z}|= s$ for $s \in \mathbb{R}$, say for $s= 1, 1/2$. Following is the code but it doesn't ...
- 123
1
vote
0
answers
108
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Why ResidueSum returns 2 different values?
I want to calculate the sum of the residues of the function
$$ \frac{\sin(\theta)\tanh(\alpha x)}{-\cos(2xL)+\cos(\theta)}, $$ for $|x| \leq1$. The function has singularities at the points $x_n^\pm = \...
- 11
2
votes
2
answers
155
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Solve an equation involving Exp over the complexes
I need some help to solve an exponential equation with Mathematica over the complexes.
This is the equation I need to solve. I've already used FullSimplify.
...
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1
vote
1
answer
150
views
How to return only the negative complex solutions of DSolve or using FindInstance?
I am using DSolve to find the solutions to two differential equations. The problem is from the first solution, I only want the solution that includes a negative ...
- 121
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0
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63
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How do I find the definite double integral of a real number over a complex region?
This is a follow-up question of How do I take the integral of a real number over a complex region?
Given that $\tau'$ is imaginary, how would you calculate the definite double integral over a complex ...
- 1,560
1
vote
1
answer
47
views
How to plot a 3D-subspace of a $\mathbb{C}^2$ curve?
I am trying to visualize the intersection of a complex curve in $\mathbb{C}^2$ with various 3D subspaces but I am not sure how to get Mathematica to do it (so far I have been able to visualize ...
- 11
1
vote
2
answers
133
views
How to simplify expressions with complex exponents?
I have the following expression
expr = -E^(-I k) Sqrt[ E^(I k) k1 k2 + E^(3 I k) k1 k2 + E^(2 I k) (k1^2 + k2^2)]
with real k, k1, and k2. I want to simplify it to
...
- 41
1
vote
1
answer
84
views
Numerical integration over Re[expr] still has $0 i$ in the result
I am using NIntegrate to evaluate the integral of the real part of some function.
...
- 179
2
votes
1
answer
136
views
2
votes
1
answer
82
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What is the status of ComplexAnalysis`BranchPoints and ComplexAnalysis`BranchCuts?
In 13.3 and I believe quite a few versions earlier I could perform
...
- 9,647
2
votes
2
answers
666
views
How do I take the integral of a real number over a complex region?
If $A$ is a constant, real number with respect to $\tau$ and $\tau$ is an imaginary number, then how would I compute the integral in Mathematica?$$f(\tau)=\int A\space d\tau$$$$\tau=ir$$$$f(r)=Ai\int ...
- 1,560
2
votes
1
answer
102
views
How to convert DifferenceRoot into a special function?
Clear["Global`*"];
f[z_] := z^(2 m) /(1+z)^m
res = SeriesCoefficient[f[z], {z, -1, -1},
Assumptions -> Element[m, PositiveIntegers]]
The result ...
- 1,943
0
votes
0
answers
90
views
Missing plot for double derivative with log
I want to plot the given code of double derivate with log function. My derivative is a function of delta P. But when I run the code, I never see the plot.
...
- 17
2
votes
0
answers
54
views
Plotting diagram of complex equation [closed]
I have this equation:
q1[t_] := -I*γ*Sin[θ] + ((γ*Cos[θ])/n^2)*(I*ξ[t]*y+z*α[t])
and the main code is the following with the definition of each parameters:
...
- 63
0
votes
1
answer
54
views
Taking the Unit Disk as a Domain for ComplexRegionPlot
I am very new to Mathematica and overall not very good with programming. I want to plot the region $|(\frac{z}{f(z)}f'(z))^{1+q}-1|<1$ where $z$ is taken from the unit disk and $0<q<1$. In ...
- 1
0
votes
0
answers
65
views
Successive multiplications using Mathematica tools
Successive rotations on the unit circle
Given an arbitrary starting point $z$ on the unit circle in the complex plane, the goal is to compute a cycle of rotations by $\pi$/2 with successive ...
- 1,050
0
votes
2
answers
120
views
Complex integral with branch cuts
I am struggling with a complex double integral with multiple branch cuts. Even the single variable complex integral I find quite complicated due to the branch cut and the special functions involved in....
- 417
2
votes
1
answer
49
views
How to simplify the following expression of a matrix with complex conjugation?
Consider the following matrix:
...
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97
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Mathematica fails to correctly recognize Hermitian matrix
I have a Hermitian matrix, but Mathematica gives 'False' when I try with HermitianMatrixQ.
...
1
vote
2
answers
105
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How to transform a complex quotient involving absolute values and conjugates into the rectangular form?
I am relatively new to Mathematica and I am still trying to figure out all the commands.
I have these three complex numbers:
And I wish to calculate the following quotient and write it in the a+bi ...
0
votes
0
answers
52
views
How to correctly represent any Power function and How to draw the image of Power function correctly? [duplicate]
Why is the result of solving this inequality incomplete?Is it a BUG in the software?
According to cvgmt's answer to the previous question, I review the function images of Power function again, and ...
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0
votes
1
answer
100
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Why is the result of solving this inequality incomplete?Is it a BUG in the software? [closed]
Solve this inequality
In[41]:= Reduce[x^2 > x^(1/3), x, Reals]
Out[41]= x > 1
Missing a solution as
x<0
There was ...
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6
votes
1
answer
156
views
Result of numerical computation representing a real physical quantity still contains a small imaginary components [closed]
I have been grappling with a physics calculation pertaining to thin film optics. I have been endeavouring to use transfer matrix calculations to determine optical reflectance. Attached below is the ...
2
votes
2
answers
59
views
Computing the continuous eigenvalues of a family of matrices
I want to compute the eigenvalues of a family of $2 \times 2$ unitary matrices $M: [0, 2 \pi] \to U(2), k \mapsto M(k)$, which is given by
\begin{align*}
M(k) = \frac{1}{2} \, \begin{pmatrix}
...
- 103
4
votes
2
answers
200
views
Simple sum runs endlessly
I tried evaluating
Sum[-(1/(2 X (-X+vF (I+y) (I T+n β))^3)),{n,-∞,∞},GenerateConditions->False]
which did not terminate. Whereas the modified sum
...
- 1,648
3
votes
3
answers
199
views
Given the u or v of the analytic function f(z), find f(z)
Given the u or v of the analytic function f(z), find f(z):
z=x+I y,
f[z]=u[x,y]+I v[x,y]
Given
$v=\left[-x+\left(x^2+y^2\right)^{...
- 1,943
1
vote
2
answers
106
views
What is the difference in plots features between analytic (holomorphic) and non-analytic functions?
We know FunctionAnalytic can check whether a complex function is analytic. But I want to use ComplexPlot3D or ...
- 1,943
2
votes
1
answer
103
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Why does Wolfram|Alpha give inconsistent result for $\int_0^\infty \frac{e^{x-x^2}-e^{-x-x^2}}{x}~dx$? [closed]
Why Wolfram gives inconsistent result for :
$$I=\int_0^\infty \frac{e^{x-x^2}-e^{-x-x^2}}{x}~dx$$
As shown in the picture below, the analytical integration gives $-i\pi$, but numerical integration ...
- 183
1
vote
0
answers
40
views
Getting hue correct in a complex argument plot [duplicate]
I'm trying to generate a picture of the phase of a complex function defined on a rectangle. I'm using Hue (which is cyclic) and DensityPlot, but it's leaving out certain "corners" of the ...
- 455
1
vote
3
answers
143
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Expanding a complex number
How does one get Mma to expand
(1/2 + I/2)/Sqrt[2]
into
1/(2 Sqrt[2]) + I/(2 Sqrt[2])
Even evaluating this second expression ...
1
vote
2
answers
110
views
Prove complex equation
Assuming $\frac{z_2-z_1}{z_3-z_1}=\frac{z_1-z_3}{z_2-z_3}$, to prove
$$
\left|z_2-z_1\right|=\left|z_3-z_1\right|=\left|z_2-z_3\right|
$$
...
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3
votes
4
answers
572
views
Prove that the modulus of a complex function is 1
Assuming that both $a$ and $b$ are complex and $| b |<1$, $| a |=1$, I want to prove that $\left|\frac{a-b}{1-a^* b}\right|=1$.
My code:
...
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1
vote
1
answer
113
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Conjugating real elements
Facing the same problem reported here: Conjugation with real elements answers are working, I thought of the following solution:
...
- 648
2
votes
2
answers
194
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Import complex numbers from a CSV file created in MATLAB
I export a simple CSV file created in MATLAB with complex numbers, such as:
0.1+2.1i,0.9+0.13i,0.02+0.3i
I would like to import it into Mathematica with the "i" recognized as the imaginary ...
- 115