Questions tagged [calculus-and-analysis]
Questions related to the calculus and analysis branches of Mathematica, including, but not limited to, limits, derivatives, integrals, series, and residues.
4,924
questions
2
votes
2
answers
113
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How to get the explicit result of this limit?
Let $f(x) = x^2+1$. I would like to compute the limit
$$
\lim_{k \to +\infty} k \int_0^1 \sin \left [ \left ( k + \frac{1}{2} \right ) \pi x \right ] f(x) \, \mathrm d x.
$$
Then I use the command
<...
- 157
2
votes
3
answers
156
views
How to make Mathematica gives the right answer for this integral $\int_{-\infty}^\infty \frac{\mathrm dx}{(a^2 - x^2)^2 + (bx)^2}$?
I'm trying to compute the following integral with Mathematica: $\int_{-\infty}^\infty \frac{\mathrm dx}{(a^2 - x^2)^2 + (bx)^2} = \pi/ab^2$. Source . Somehow, even if I assume $a>0$ and $b>0$, ...
- 123
1
vote
1
answer
73
views
Is the given $\lim_{x\to\infty} f(x)=0$ correct while we see the argument of cosine may become close to $\frac{\pi}2$?
I have this function $f(x)$ for integer values of $x>0$.
$$f(x)=\left\lvert\frac{4}{\left(3 x^2\right) \cos \left(\sqrt{10} x\pi\right)}\right\rvert \quad \text{for}\quad x \in\mathbb N$$
I use ...
- 699
0
votes
1
answer
79
views
Mathematica kernel dies
When I run the following code, the kernel dies. I have tried making a new kernel, but that also does not work. The memory usage never crosses 33% of the total RAM available. I have no clue how to fix ...
- 9
3
votes
0
answers
90
views
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.
...
- 24.5k
5
votes
1
answer
125
views
Strange result simplifying higher order BesselJ
Consider the following integral:
$$\int_0^1 Z_n^m(r)\ J_m(\rho r) r dr$$.
The solution of this should contain a single Bessel function:
$$(-1)^{(m-n)/2}\ J_{n+1} (\rho)/\rho$$ (see https://www.osti....
2
votes
1
answer
125
views
Graph for derivative function
Please help. I can't display a graph for the derivative of a function.
...
- 109
1
vote
0
answers
68
views
How do I integrate with respect to one variable when an integrand has many variables?
I am trying to integrate the following integrand:
(μ*a*curr/4 π)*((x - a*Cos[ϕ] + y - a*Sin[ϕ])/
((x - a*Cos[ϕ])^2 + (y - a*Sin[ϕ])^2)^(3/2))
with the following ...
- 11
2
votes
1
answer
84
views
How to make Assumptions in IntegrateChangeVariables?
There is a new useful command IntegrateChangeVariables since 13.1.
The command has the Assumptions option which is very poorly ...
- 24.5k
6
votes
1
answer
488
views
Symbolic integration of a sigmoid function
I'd like to obtain a symbolic result for the following integral:
Sigmoid[x_] := 1/(1 + Exp[-x])
Integrate[Sigmoid[x]*Log[Sigmoid[x*w]], {x, -Infinity, Infinity}]
...
- 163
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
0
votes
0
answers
64
views
Help implementing Lagrange inversion
I'm attempting to find the coefficients of generating functions using Lagrange inversion. I am finding the series coefficients with:
...
2
votes
2
answers
230
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The center of mass of a semiellipsoid
I am trying to find the center of mass of a semiellipsoid using cylindrical coordinates.
r^2/a^2+z^2/b^2 < 1, z < 0 the density = 1.
I know that the center of mass is (1/M int,int,int rcos(theta)...
- 31
9
votes
5
answers
3k
views
Digits of Pi in colored spiral
In How to make the digits of π go around in a spiral like this?
it is described how to plot pi in a spiralform (in my case as binary number):
...
- 93
0
votes
1
answer
50
views
Plotting an integral with solving for a specific constant [closed]
Given an Integral I(t) which is a function of a constant tc, this constant is determined with the condition I(tc)=0, how can I add this condition?
- 117
0
votes
1
answer
51
views
Numerical Integration without giving all values [closed]
Is there a way to solve expressions numerically, while keeping other variables.
For example, I want to Integrate the expression below for $\theta p,\theta q$ while keeping other things as it is. The ...
- 9
2
votes
1
answer
48
views
How to linearize these terms in many different variables? [duplicate]
How to linearize these terms in q[t] and p[t,y]
...
- 213
0
votes
2
answers
76
views
Integrate with `Max` and `Min` under assumptions runs forever
I'm trying to solve an integral under certain conditions on the parameters. The following is my code.
...
- 981
1
vote
1
answer
52
views
Leading significant digits
Given that the initial area in a Sierpinski triangle is A_0 and the subsequent areas
$$(A_n) =A_0 (3/4)^n$$ ,
what is the formula to calculate the leading ...
0
votes
1
answer
75
views
What does "{True,True}" mean in the result of IntegrateChangeVariables?
I want to change variables in the following integral:
Integrate[q*D[z[x, y],x]^2, {x, -∞, ∞}, {y, -∞, ∞}]
as follows:
...
- 38.7k
0
votes
0
answers
98
views
Integral of complex exponential with assumptions
I feel confident this problem has been posed before but alas I cannot find it.
...
- 40k
2
votes
3
answers
144
views
Find Generalized Series with Symbolic Variable
CoefficientList[Series[Exp[x], {x, a, 3}], x]
Gives the following expression,
$$
\left\{-\frac{1}{6} e^a a^3+\frac{e^a a^2}{2}-e^a a+e^a,\frac{e^a a^2}{2}-e^a a+e^...
- 123
0
votes
0
answers
50
views
Express the double integral in terms of the function values at discrete nodes
Consider the following integral:
$$
\mathcal{I}[p_{1},t] = \int \limits_{0}^{p_{1}}dp_{2}\int \limits_{0}^{p_{2}}dp_{3} \ \text{integrand}
$$
where
...
- 4,731
0
votes
1
answer
63
views
How to linearize this equation in two different variables?
How to linearize this equation in two different variables p[t] and q[t,y]:
...
- 213
1
vote
1
answer
54
views
First order differential equation with two conditions
I want to produce all possible solutions for this equation:
...
- 153
0
votes
0
answers
48
views
Limit giving indeterminate result
I have a function $r_h(v)$ given by,
$$r_h (v) = \left(\frac{m_0}{2 g} e^{-g v_f} \left(-1+e^{-g(v - v_f)} \right) \right)^{1/5}$$
where $m_0$ and $g$ are just numbers. I want to take the limits of ...
- 691
0
votes
0
answers
106
views
Integrate function differing from RUBI's Int function in version 13.3.0
I had Mathematica Version 12.3.1 Student Edition and Integrate and RUBI's Int functions produced the same answers for my purposes with good results. I updated my version to 13.3.0 Student Edition and ...
- 9
1
vote
1
answer
100
views
Plot after applying NIntegrate
I'm trying to integrate and graph $I$ vs $b$ the following integral (eq. 24 in this paper) in wolfram mathematica
$$ I = \int_{2}^{\infty} \frac{g_i^3 \mathcal{K}_t \mathrm{~d} r}{r^2\left|\mathcal{K}...
- 95
0
votes
1
answer
85
views
0
votes
1
answer
93
views
Change of variable of a numerically solved integral
I have the following function that numerically integrates a function on the interval $\left[ 2, 100 \right]$
...
- 95
1
vote
0
answers
81
views
Intensity pattern [closed]
I'm trying to get the following intensity pattern from this paper
In the Newtonian case, the equations for the intensity (eq. 9 and 10 in the article), I put them in Mathematica with the change $b^2 =...
- 95
1
vote
1
answer
88
views
Definite integral of real function outputting complex values
I'm trying to integrate the function $f(\omega)$ from $-\infty$ to $\infty$, where
$$
f(\omega) = \begin{cases}
\frac{(1-2 \omega )^2}{64 \cos (\tau (-\omega +\phi
+1))-80} & 0<\omega \leq ...
- 111
0
votes
1
answer
52
views
How to chose appropriate finite limits for NIntegrate?
I want to NIntegrate a function using finite limits so it gives a result as close as possible to a result which I would get making the limit ...
- 55
1
vote
1
answer
257
views
Plot solution of an integral with parameter
In this article, they solve the following integral
$$ \mathcal{I}_{o b s}\left(\nu_0\right)=\int_{\gamma_i} \frac{(A(r))^{3 / 2}}{r^2} \sqrt{\frac{1}{A(r)}+r^2\left(\frac{d \varphi}{d r}\right)^2} d r ...
- 95
2
votes
2
answers
85
views
How to prevent evaluation without simplification?
I am trying to compute a sequence of terms, which have a form similar to $g(z)$ at different points $z$. For $z$ close to 0, I have observed that the terms are evaluated as 0/0 (indeterminate form) ...
- 121
8
votes
3
answers
3k
views
Why doesn't Mathematica recognise two integrals as equal?
The following expression
$$\int_{0}^{t} f(t_0) \,dt_0 - \int_{0}^{t} f(t_1) \,dt_1 = 0$$
is zero because it is just a change of integration variable.
Why doesn't Mathematica give zero in this case?
<...
- 83
4
votes
1
answer
185
views
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
1
vote
2
answers
52
views
How to find the roots of an equation for different wave numbers?
I have a equation nn which is a function of t and k. I want to find the maximum of the ...
- 349
1
vote
2
answers
186
views
Sum $\sum _{k=0}^{\infty } \frac{(-z)^k}{\text{IteratedLog}(k+2)}$ [closed]
The following result can be obtained by means of succession acceleration. Can this result also be calculated in Mathematica?
$$\sum _{k=0}^{\infty } \frac{(-z)^k}{\text{IteratedLog}(k+2)}=\frac{z+2}{2 ...
- 43
0
votes
1
answer
133
views
0
votes
0
answers
59
views
Plotting list of points to a special graph
Given Mathematica code of eigenvalues in different variable-paths , I want to add experimental data in the path Gamma-X ; X-Gamma ; and M-Gamma .
Here is the code :
...
- 117
2
votes
0
answers
273
views
Symbolic integral with UnitStep functions
I want to symbolically evaluate the following integral (with real positive parameters L1, L2, ...
1
vote
1
answer
79
views
Minimization of an implicitly specified function
exp - an experimental data, where expT - temperature, expChi - magnetic susceptibility
I ...
- 1,753
0
votes
1
answer
58
views
Syntax error trying to rewrite expression using rules
I want to use Mathematica to calculate derivatives of some long, complicated expressions. I have an integral with integrand $I_*$ which is a functional that depends on $h(x,t)$ and $\phi(x,t)$. The ...
- 221
0
votes
1
answer
72
views
Divergent series $\sum _{k=0}^{\infty } (2 k+1)^{3 k+1} z^k$ and LambertW function [closed]
I can get values from the following function,
$$\sum _{k=0}^{\infty } (2 k+1)^{3 k+1} z^k $$
could you tell me how you could check those values using the Lambert function in integral form or if you ...
- 43
1
vote
1
answer
152
views
Why does Mathematica fail to compute $\int_{0}^{\infty} \sin(ax^2) dx$ whereas Wolfram Alpha seems to be able to do it? [closed]
I was symbolically computing
$$ \int_{0}^{\infty} \sin(ax^2) dx $$
Via the command:
Integrate[Sin[a*x^2], {x, 0, Infinity}]
But it seems to fail with mathematica, ...
- 133
0
votes
1
answer
51
views
Different integral values for analytically same integrals
Consider a function $L$ with parameters $h=0.1,\ D=1.5$
$L(m,x):=e^{-\frac{2\ (\frac{x}{h}-m)^{2}}{D}}$.
While the integrals $\int_{-\infty}^{+\infty}L(0,x)dx$ and $\int_{-\infty}^{+\infty}L(40,x)dx$ ...
- 3
0
votes
3
answers
137
views
Finding the area enclosed by two curves
I am trying to find the area enclosed between two curves (i.e., two datasets).
The challenge is to estimate the cross-sectional area.
as indicated in blue.
The first dataset represents the bathymetry ...
- 587
3
votes
6
answers
449
views
Galactic rotation speed
Rotation speed vc[R] in reduced units for M33 looks like
0.342331 Sqrt[R^2/(0.728117 + R^2)^0.87955]
Knowing ...
- 193
2
votes
0
answers
143
views
Finite sum of complex exponentials giving wrong result [closed]
I was playing with the following sum:
$$ \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