Questions related to the calculus and analysis branches of Mathematica, including, but not limited to, limits, derivatives, integrals, series, and residues.

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1
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2answers
219 views

How to make Mathematica use the chain rule?

Lets say I have the following PDE: $$x^2 u_{xx} - u_{yy} + u_y = 0$$ And I have the following change of variables: $$ s(x,y) = x e^y \, \, \, , \, \, t(x,y) = x e^{-y}$$ How can I use Mathematica ...
0
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2answers
44 views

Problem with using the chain rule [closed]

I have the functions w[x_] := x*Cosh[x] and x[a_] := n/k*a. I take the derivative ...
0
votes
0answers
60 views

Evaluating an indefinite integral at the boundaries

I have the following expression i need to integrate over $\alpha$ from 0 to 1 and $x$ from $x1$ to $\infty$ ...
0
votes
1answer
86 views

Help with Computation-heavy triple integration

I'm Trying to use Mathematica to symbolically integrate a function with respect to x, y, and z. For some reason, this computation either takes about, 60s and returns the input, or takes many hours ...
1
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0answers
84 views

Help wanted in evaluating an integral

Trying to evaluate an integral. To be specific, I'm trying to find the expectation value of $x^2$ for the ground state of hydrogen. Here is the code I used: ...
0
votes
1answer
44 views

Derivative of the composition of two functions

Background, this seems to work (This is manually transcribed from the program, beware of errors): J1[f_, i_] := DifferenceDelta[f[i], i]^2 D[J1[f + t v, i], t] ...
1
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0answers
54 views

Working with abstract vectors

I often need to compute derivatives or integrals involving N-dimensional vectors (where the dimension could be equal to 2 or 3 but is not particularly relevant for the sake of the derivation). The ...
0
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2answers
90 views

How to calculate an improper integral

How can I find the range of the parameter a such that the improper integral Integrate[(x^(a - 1))/(1 + x), {x, 0, ∞}] ...
8
votes
1answer
189 views

Why is this infinite series wrongly computed by Mathematica?

Could you let me know if Mathematica (newer versions) is able to correctly compute this one? Sum[(-1)^(n + 1) Cos[3^n x]^3/3^n, {n, 1, Infinity}]
0
votes
1answer
63 views

Integration of linked PDFs over probability simplex

I'm trying to integrate the following expression: f[a1,p1+p2+p3+p4]*f[a2,p2+p3+p4]*f[a3,p3+p4]*f[a4,p4] Where ...
0
votes
1answer
75 views

Numerical vs Symbolic Integration: Loss of precision

I am trying to integrate the following expression over the time interval $0\leq t \leq \text{period}$. ...
1
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4answers
318 views

How to calculate this integral?

I am trying to Integrate the following Integral : $\int_1^{\infty } \dfrac{\left(x^2-1\right)^{13/2} e^{-ax} }{x^{10}} \, dx \,\, \,\,\,\,\,\,\,\,\,\,\,\,(a=\textrm{real>0})$ Mathematica didn't ...
2
votes
2answers
259 views

How to compute this triple integral? [closed]

In Mathematica how can I compute this integral:$$ \iiint_{D}\sqrt{(1-9z^2)(1-4y^2-9z^2)}\,dx\,dy\,dz$$ where D is the domain: $$D: x^2 +4y^2+9z^2\le1$$ Please I need help!!!
18
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2answers
269 views

Negative probability?

I am trying to get the sum of the squares of seven random variables, all uniformly distributed. This is what I tried. ...
2
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0answers
55 views

Integration of UnitStep or HeavisideTheta

I am using Mathematica 9. If I evaluate Integrate[Piecewise[{{1 - r, r > 0 && r < R}}, 0], r] then I get ...
0
votes
2answers
133 views

Numeric mixed derivatives

In short I need NumericCalculus`ND extended to mixed partial derivatives. This can be done by nesting them, but care has to be taken to evaluate underling ND only when the parameters become numeric. ...
9
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3answers
264 views

Correct way to integrate a certain function

I want to integrate $$ \int_{0}^{2\pi} dt_1 \frac{a + b*\cos(t_1 - t_2)}{c + d*\cos(t_1 - t_2)} $$ where $a, b, c, d, t_2$ are real numbers and $c + d > 0$ & $0 \leq t_2 \leq 2\pi$. I used ...
2
votes
2answers
82 views

Graphical solution of a a two-variable system containing an equation and an inequality

I'm currently studying atmospheric-gravity waves and their dispersion relations. I want to have look at values of $(x,y)$ that solve the following system : \begin{equation}\label{}(1) \left\{ ...
0
votes
0answers
49 views

Complex integrals and residues

I have a question. I try to integrate this function: Integrate[Exp[I k z]*Exp[-Sqrt[k^2 - A^2]x]*Exp[-k^2*L^2/4]/Sqrt[k^2 - A^2], {k, -Infinity, -A}] But ...
0
votes
1answer
34 views

Defining arbitrary order derivatives function [duplicate]

I have a function $\phi(t)$ and I want to define a function $\phi D(t,r):=\phi^{(r)}(t)$, in other words evaluate the $r$'th derivative of $\phi$ at $t$. The naive code that I have is: ...
17
votes
2answers
824 views

Bug in mathematica analytic integration?

I found Mathematica provides me a wrong answer for a relatively simple analytically solvable integral: ...
11
votes
3answers
416 views

Determine frequency of oscillations

I am wondering how I could determine the frequency of oscillations of a differential model equation? How could I find the frequency from this example given in Mathematica Documentation: ...
1
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1answer
76 views

Two graphs in the same plot [closed]

I have the following plots for two graphs: ...
2
votes
3answers
84 views

Converting HeavisideTheta[]s and Sign[]s functions to a single Piecewise[]

I have the product of some Heaviside functions or sign functions, a simple example could be: HeavisideTheta[q^2 - 4]*Sign[q - 3] Another slightly more ...
7
votes
1answer
627 views

Is Mathematica really getting this limit wrong?

I'm trying to calculate: $$ \lim_{\beta \to \infty} \tanh \left( \beta A \right) = \mathrm{sgn} \left( A \right) $$ for $A \in \mathbb{R}$, $\mathrm{sgn} \left( \# \right)$ being the sign function. ...
0
votes
0answers
77 views

A power series expansion

Consider the function, $f(z) = z tanh(\pi z) log (z^2 + a^2)$ for some $a>0$. Now I am considering 3 different situations, $z = i(n+0.5) - i\epsilon + \delta - it$ for $n \in \mathbb{Z}$ and ...
0
votes
1answer
94 views

About doing an sum

I want to explain this summation to Mathematica, For $a >0$ define $n_u, n_d \in \mathbb{Z}$ such that if $\{ a \} \leq 0.5$ then $n_u = [a]-1$ and $n_d = - [a]$ and if $\{ a \} > 0.5$ then ...
6
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1answer
111 views

Is it possible to circumvent a bug inside SeriesCoefficient?

As far as I can tell, there seems to be a bug in SeriesCoefficient: ...
2
votes
1answer
94 views

Prove (or check) an expression is positive given constraints on variables?

I'm trying to figure out whether an expression is always positive given positive parameters. When the expression is complicated, I can't do this by eye. Is there any way to make Mathematica prove ...
1
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0answers
115 views

Derivative of the Absolute Value Function in Mathematica looks very different from the actual answer

Derivative of the Abs Function in Mathematica looks very different than the actual answer. Why? How to correct this? For example I entered ...
0
votes
1answer
127 views

Computing the volume and the curved surface area

I'm trying to compute this function $y=\frac{1}{2}e^x+\frac{1}{12}e^{-3x}$ to get the volume of the solid of revolution when the area under the curve is revolved once around the x axis. And than get ...
0
votes
1answer
74 views

How to the take derivatives of a parametric function? [closed]

I have a parametric curve of the form $x(t)=2t^3+3t^2-12t,\ y(t)=2t^3+3t^2+1$ How do I take the derivative of that function in Mathematica? Is there a certain function I should use?
0
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0answers
69 views

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 ...
4
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1answer
85 views

When to use GenerateConditions -> True

Many functions, usuallly those involving integration, take a GenerateConditions option which often defaults to False, or at ...
2
votes
1answer
162 views

Integrate does not accept my integrand expression

I want to calculate the symbolic integral, but I could not obtain the final results from Mathematica. ...
1
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2answers
142 views

Making mathematica do regulated integrals

Consider the integration, $\int _0 ^\infty dx\ x \tanh( \pi x) \sqrt {x^2 + a^2 } $ where $a$ is a real number. This integral is divergent. We note that an asymptotic expansion of the integrand ...
0
votes
0answers
48 views

problem solving a simple integral equation

I'm looking for a function q(x) so that integral( q(t) , t , x, x+2*pi/3) == a*sin(x)+b Could anyone help me on this one? Thx in advance
0
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0answers
60 views

Computing residues of matrix valued functions symbolically

I have a somewhat specific application in mind, but I've run into this kind of problem a few times and never found an elegant solution. Is there an efficient way to get Mathematica to recognize that ...
3
votes
2answers
235 views

How can I calculate the limit without using the L'Hopital's rule

I need to prove this limit without using the L'Hopital's rule: $$\lim_{x\to 0} \frac{(1+a\,x)^{1/4} - (1+b\,x)^{1/4}}{x} = \frac{a-b}{4}$$ How can I do it in Mathematica?
1
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2answers
75 views

Integrating a half-normal pdf

The pdf of a half normal distribution is: $\frac{\sqrt{2}}{\sigma\sqrt{\pi}}\exp\left(-\frac{x^2}{2\sigma^2}\right)$, $x>0$ ...
1
vote
1answer
74 views

RegionPlot of a Double integration [closed]

Hey guys I have a problem where I want to sketch the region of integration: Integrate[x Cos[y^2], {x, 0, 3}, {y, x^2, 9}] I know I have to use ...
3
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0answers
75 views

Derivative of generating function (Example from documentation)

In the documentation for GeneratingFunction, the following example is given under Examples -> Properties & Relations -> Derivative: ...
3
votes
1answer
82 views

Why does Mathematica list the variables in a double integral backwards? [closed]

I am an advanced novice Mathematica user and have done a fair amount of single-variable calculus things with it. This semester I am using Mathematica for multi-variate calculus and when I tried to ...
1
vote
2answers
98 views

Using Mathematica to derive the PDF of the normal distribution

I am trying to use Mathematica, in this case in a well known and rather simple list of equations, but I get stuck whit Mathematica. Since I am just starting to use Mathematica for mathematical ...
0
votes
1answer
54 views

NIntegrate and NumericQ [closed]

I need to do numerical integration --> differentiation --> numerical integration. I tried to apply the technique described in the accepted answer to a similar question: ...
0
votes
0answers
30 views

Is it possible to obtain vector differentiation results in terms of vectors rather than components? [duplicate]

When doing certain operations on vectors in Mathematica it is quite desirable to convert the results back into coordinate-free notation. Unfortunately, after going through the Documentation Center and ...
1
vote
1answer
218 views

How do I get all possible solutions in an underdetermined system? [duplicate]

I have two problems which I'd like to solve with Mathematica. If I have a system of two equations with three unknowns, how can I get to list all possible solutions for the unknowns? Here is what I ...
6
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0answers
149 views

Incorrect evaluation for Thue-Morse signed harmonic series

I would like to evaluate $$s = 1 - \frac{1}{2} - \frac{1}{3} + \frac{1}{4} - \frac{1}{5} + \frac{1}{6}+\frac{1}{7}-\frac{1}{8} - ... + \frac{(-1)^{\textrm{binary digit sum}(n-1)}}{n} + ... $$ where ...
6
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1answer
362 views

What is the mathematical meaning behind D[f]?

y = a + b x; I can understand this output of the ordinary differentiation of y w.r.t. x ...
0
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0answers
77 views

Numerical Differentiation of an interpolated list.

I have a synthetic spectrum with a wavelength range of .3 to .7 microns. I have interpolated this spectrum for re-sampling purposes. However, I am also attempting to calculate the first and second ...