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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About how Mathematica understands the branchcuts of the complex logarithm [Part 3]

One understands that for the function $\log(z^2+a^2)$ Mathematica implicitly puts the branch cuts to be starting at $\pm ia$ and going up/down respectively on the imaginary axis. The phase of this ...
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0answers
34 views

Conjugate of InterpolatingFunction?

From NDSolve, I got this: {{a1 -> InterpolatingFunction[{{0., 40.}}, "<>"]}} What would be a sensible approach to ...
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0answers
58 views

Deformation of pictures [on hold]

If I get a picture, and imagine it as a complex plane, then given a function how to visualize the resulting deformed picture? Put it in this way: choose the left bottom vertex of the picture as the ...
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3answers
147 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: ...
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1answer
62 views

Series expansion of a complex function

How do I expand a function $f(z)$ in a particular region? For example, how would I expand $f(z)=(z^2-3z+2)^{-1}$ in the region $0<|z-1|<1.$? I believe this can be done by the binomial theorem. ...
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6answers
4k 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 ...
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4answers
497 views

Moving the location of the branch cut in Mathematica

According to the documentation, Mathematica chooses the branch cut for $\log(z)$ to lie along the negative real axis. It it possible to change this so that it lies along the positive axis or elsewhere ...
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48 views
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1answer
413 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 ...
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2answers
1k 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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0answers
144 views

Fastest way to create a NxNxNxN dimensional matrix

I have a function that outputs a complex number Func[x_,y_]:= blah blah; I have constructed a table of all possible values of this function. ...
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0answers
53 views

How does Mathematica understand branchcuts of the complex logarithm? [Part 2]

I observe that the answer generated by the first integration is not in any simple way related to the next two answers. (some simple jump of i 2\pi across the branchcut of the logarithm is not helping ...
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1answer
40 views
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0answers
32 views

Chopping pure Imaginary numbers [duplicate]

I would like to be able to Chop complex numbers and display only the non-zeroed real or imaginary part when applicable. Chop alone works just fine if the non-negligible part is only the real part. ...
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1answer
117 views

I integrate a real function, I want a real result [closed]

I want to count some integral in Mathematica (actually, check that this integral is zero...) I arrive to an answer which has a zero real part but has some imaginary part! Why? This could be a mistake ...
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1answer
77 views

A question about using Reduce [closed]

Can someone help decipher what is the meaning of this computation that Mathematica has done? ...
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1answer
102 views

How to do this complex integration on the real line?

$m, r$ are parameters in the following integral: Integrate[z Exp[I z r]/Sqrt[z^2 + m^2], {z, -∞, ∞}] How to do this integration directly? The result should be ...
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1answer
44 views

Determine types of input, output and internal variables in compiled functions

I am trying to speedup the computation of a longer function using Compile. For illustration consider the following short example: ...
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1answer
48 views

Simplifying complex expressions

I'm trying to work with complexes and I want Mathematica to know that all variables are real unless I explicitly assigned the variable with an I so I wrote: ...
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2answers
55 views

How to find the complex number has least and greatest modul satisfying a given condition?

Let be given the complex number $z$ satisfying the condition $|z-2+2i|=2\sqrt{2}$. I want to find the complex numbers $z$ so that their modul obtain least and greatest value. I tried. Put $z = x + y ...
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1answer
54 views

Complex LogIntegral error

Going through Derbyshire's Prime Obsession & trying to take LogIntegral of 20^ZetaZero[1] & comes up with a value of ...
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0answers
31 views

Check if a (possibly nested) list is real [duplicate]

I have a very large matrix of numbers. I want to check if all its entries are real. At the moment I am using realQ = And @@ Thread[Flatten[Im @ #] == 0] & Is ...
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1answer
202 views

How to get rid of all complex numbers and functions?

For almost esthetical reasons, I always have been fascinated by infinite sums of series and I always wonder if theyhave a closed form. A couple of days ago, I found on MSE a question related to the ...
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2answers
74 views

Plot isn't outputting a plot [closed]

I am trying to plot the magnitude complex function (f) with respect to w. For some reason, Plot is just displaying a set of axes, with no points. Any Idea what's going on here? ...
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0answers
33 views

Plotting a complex valued function in mathematica [duplicate]

How can I plot a complex valued function like $f(z)=z^2, z=x+Iy$, in Mathematica? Can anyone help?
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1answer
95 views

Plotting discrete points on a phase plot

I'm trying to generate a phase plot of a sinusoidal voltage at discrete points using the following code: ...
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1answer
55 views

decomposing a complex equation

I have an equation such as: a*z^2 + b*(z - c)^2 == d where z is a complex variable and ...
43
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4answers
4k 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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1answer
779 views

How can I get the solution of complicated implicit function?

The question is what is the method to solve the implicit function has real and imaginary number. For example, The function is $$F(x,y)=(-I*x + 2*y^2)^2 + x^2 - 4*y^4*Sqrt[1 - I*x/(y^2)]$$ Although ...
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2answers
121 views

Complex limit gives wrong answer

I'm evaluating $$\lim_{z\to e^{i \pi/2 n}} \frac{z-e^{i \pi/2 n}}{z^{2 n}+1}$$ with Mathematica 9.0.1: ...
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1answer
143 views

Simple way to plot a complex valued function of one real variable

How can I plot a complex valued function of one real variable, $f(t)$, as $$x=-\Re f(t),\ z=\Im\ f(t),\ y=t \space?$$ My aim is a 3-dimensional path, treating the complex valued function as if it ...
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0answers
66 views

Alternate method of conjugation using replaceall

Mathematica really struggles to simplify complex conjugates of complicated expressions - even given assumptions for every symbol in the expression being real. Often times I've found that I can ...
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1answer
76 views

Solving Complex Equation Over Reals

I want to solve the equation $\frac{abi}{a+bi}=4-2i$, where $a$ and $b$ are real numbers. I know from hand-solving the answer is $a=5$, $b=-10$. How do I get Mathematica to tell me this? I tried: ...
4
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1answer
154 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?
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2answers
176 views
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2answers
688 views

How do I find line integrals?

For example, how can I calculate $$\int_{\left | z \right |=1}\frac{dz}{z}$$ I know that the answer is $2\pi i$ but how do I do it using Mathematica?
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2answers
390 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: $$\oint_{\mid z \mid =2}\frac{1-e^z+z}{z^3 (z-1)^2}dz$$
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1answer
111 views

Bifurcation with a system of equations

I'm trying to plot a flip bifurcation diagram for a dynamical system of equations as follows: x'[t] == v[t] v'[t] == x[t] - A x[t]^3 + R*Cos[\[Omega]*t] - B v[t] ...
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2answers
127 views

Limit of a complex function

Can someone show me how to find the limit of a complex function? Example: z1 = 3 + 4*I some_function[z] = z * z1 Set-up: ...
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0answers
69 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: ...
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1answer
145 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: ...
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0answers
67 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 ...
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2answers
282 views

Export complex number to excel or txt file

I'm doing some calculations on complex numbers in Mathematica. I can use Export to generate a file (*.CSV or *.dat, or anything else) of complex numbers. If I want to use excel, origin or Matlab to ...
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3answers
985 views

How can I convert a complex number a+b I to the exponent form A Exp(I phi)?

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 ...
3
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1answer
254 views

Plotting multiple lists of complex numbers

The user #Nasser showed an elegant way to plot complex numbers: ...
3
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1answer
157 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 function vanishes on the line: ...
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2answers
129 views

How to plot complex values given by Solve

Update Solve[N[Table[BernoulliB[n, z], {n, 10, 10}] == 0]] ...
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1answer
56 views

Substitute complex functions into complicated polynomials

Mathematica has some strange way of sorting terms. It seems that it is using a canonical sort on expressions, e.g. $\mathbb{i} \sin[x] \cos[y] \cos[z]$ is returned as $\mathbb{i} \cos[y] \cos[z] ...
2
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0answers
159 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 ...