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.

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Strange result simplifying higher order BesselJ [duplicate]

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....
AstronomyGeek's user avatar
2 votes
1 answer
130 views

Graph for derivative function

Please help. I can't display a graph for the derivative of a function. ...
Vladimir's user avatar
  • 161
1 vote
0 answers
69 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 ...
Micah's user avatar
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1 vote
1 answer
96 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 ...
user64494's user avatar
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7 votes
2 answers
556 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}] ...
Carlo Wood's user avatar
1 vote
0 answers
39 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: ...
gaoqiang's user avatar
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0 votes
0 answers
67 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: ...
Sean Svihla's user avatar
2 votes
2 answers
236 views

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)...
Mike Gotier's user avatar
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): ...
ralf_7's user avatar
  • 93
0 votes
1 answer
51 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?
Med Ch's user avatar
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2 votes
1 answer
51 views

How to linearize these terms in many different variables? [duplicate]

How to linearize these terms in q[t] and p[t,y] ...
Dr. phy's user avatar
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1 vote
2 answers
84 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. ...
ppp's user avatar
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1 vote
1 answer
53 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 ...
Rubens Vilhena Fonseca's user avatar
2 votes
1 answer
142 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: ...
Alexei Boulbitch's user avatar
0 votes
0 answers
100 views

Integral of complex exponential with assumptions

I feel confident this problem has been posed before but alas I cannot find it. ...
David G. Stork's user avatar
2 votes
3 answers
153 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^...
Torkoal's user avatar
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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 ...
John Taylor's user avatar
  • 5,407
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]: ...
Dr. phy's user avatar
  • 285
1 vote
1 answer
54 views

First order differential equation with two conditions

I want to produce all possible solutions for this equation: ...
Mathecis's user avatar
  • 153
0 votes
0 answers
52 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 ...
mathemania's user avatar
0 votes
0 answers
107 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 ...
Tom00496's user avatar
1 vote
1 answer
104 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}...
Soliton-104's user avatar
0 votes
1 answer
92 views

Incorrect results for Integrate[Cosh[x]/(Sinh[x] + Cosh[x]), x] [closed]

Input: ...
Boris's user avatar
  • 1
0 votes
1 answer
97 views

Change of variable of a numerically solved integral

I have the following function that numerically integrates a function on the interval $\left[ 2, 100 \right]$ ...
Soliton-104's user avatar
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 =...
Soliton-104's user avatar
1 vote
1 answer
91 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 ...
peep's user avatar
  • 111
0 votes
1 answer
53 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 ...
Patrycja's user avatar
1 vote
1 answer
260 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 ...
Soliton-104's user avatar
2 votes
2 answers
87 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) ...
sai's user avatar
  • 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? <...
Luca's user avatar
  • 83
5 votes
1 answer
215 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 ...
Ring's user avatar
  • 137
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 ...
Lohrasb's user avatar
  • 359
1 vote
2 answers
193 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 ...
Capeamate's user avatar
0 votes
1 answer
138 views

How to speed up this integration?

I try to evaluate this integral: ...
Dr. phy's user avatar
  • 285
0 votes
0 answers
62 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 : ...
Med Ch's user avatar
  • 117
2 votes
0 answers
276 views

Symbolic integral with UnitStep functions

I want to symbolically evaluate the following integral (with real positive parameters L1, L2, ...
Ruth Murphy's user avatar
1 vote
1 answer
82 views

Minimization of an implicitly specified function

exp - an experimental data, where expT - temperature, expChi - magnetic susceptibility I ...
Mam Mam's user avatar
  • 1,773
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 ...
Mjoseph's user avatar
  • 221
0 votes
1 answer
73 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 ...
Capeamate's user avatar
1 vote
1 answer
155 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, ...
Sidharth Ghoshal's user avatar
0 votes
1 answer
52 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 $\int_{-\infty}^{+\infty}L(40,x)dx$ ...
chintan's user avatar
0 votes
3 answers
146 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 ...
Mehmet's user avatar
  • 587
3 votes
6 answers
455 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 ...
Rémy Galli's user avatar
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. ...
Sidharth Ghoshal's user avatar
2 votes
2 answers
95 views

How can I find maximums of a function without Table and FindPeaks?

I have the following equation $$\pi_{xx} = \exp(-2 k_x^2 \sin^2(\frac{t}{2})) k_x^2 \theta(t) \times \bigg[ -\frac{\sin(2t)}{2} - \cos(t) \sin(t) (1 - k_x^2 \sin^2(\frac{t}{2})) + 2 ...
Lohrasb's user avatar
  • 359
1 vote
2 answers
85 views

How can Mathematica merge $(1/t)^{1/3} t^{17/3}$ in an expression?

(1/t)^(1/3) t^(17/3) How to let Mathematica simplify it?
dessert's user avatar
  • 11
1 vote
3 answers
209 views

Integration giving unknown messages

I am trying to integrate the following Integrate[(Log[1 - t] t^(-(1/2) + eps))/(1 + t),{t,0,1}] To simplify this I am doing a further transformation ...
BabaYaga's user avatar
  • 1,746
0 votes
2 answers
85 views

How to find the maximum of an equation based on a variable? [duplicate]

I have the following periodic function, as a function of K and t. I need to know what is the maximum of each recurrence as accurate as possible for k={10,20,40,60,80} The equation is as follows, $pixx ...
Lohrasb's user avatar
  • 359
2 votes
0 answers
108 views

Why does Area produce finite values for this area? [closed]

Let us consider in 13.3.1 on Windows 10 the region reg = ImplicitRegion[{x*y >= 8, y >= 8 x^3, y <= 27}, {x, y}]; and its area ...
user64494's user avatar
  • 25.7k
0 votes
0 answers
21 views

The relationship between dependent decision variables in a function

Given objective function f(x1,x2,x3) in a optimization problem, and don't know if all these variables are independent or not. How to know if dx2/dx3 is greater/less than/equal to zero. I could
Doublehappy Tough's user avatar