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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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16 views

Why does Mathematica miss some solutions for my integral?

When computing the integral of two orthogonal sine functions $\qquad \sin(n\,\pi\,x)\,\sin(m\,\pi\,x)$, where $n,\,m$ are integers > 0 Mathematica seems to miss the solution where $n=m$. I don't ...
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9 views

NIntegrate a Triple Integral… Different Integration Strategies

I'm trying to evaluate a triple integral using NIntegrate. It takes the form: \begin{align} \int_{\omega_0-5\delta}^{\omega_0+5\delta} \mathrm{d}\omega \int_{\omega_0-5\delta}^{\omega_0+5\delta}\...
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1answer
61 views

How can I inform Mathematica of an identity concerning Bessel functions?

I am doing some analytical work that includes the integral of $e^{i(n\, t - x \sin t)}$. I know the result of this integral is a Bessel function. $$J_n(x)=\frac1{2\pi}\int_{-\pi}^\pi e^{i(x\sin\tau-n\...
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67 views

Minimal Surface of Revolution - Integrate Challenge

I'm trying to use Mathematica to go from equation (11) to equation (12) in this example of a Minimal Surface of Revolution. There is a MMA notebook on that link, but it only shows how to find the ...
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35 views

Graphing f '(x) from the graph of f(x) [on hold]

Let f(x) = (x^2-4)/(x^2+1). a) Use the appropriate Manipulate[ ] command from the example in this lab to graph f '(x) from the graph of f(x) using - 3 < x < 3 and a range of -5 < y &...
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35 views

Why am I getting Ceiling[j] after differentiating when j is an integer?

I'm teaching myself Mathematica and am trying to compute the following derivative $\frac{d}{dx_j}\sum_{i=1}^{n}x_i^i$ which should give $j\times x_j^{j-1}$. However, typing the equivalent in ...
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75 views

A possible bug for Limit

I was trying to calculate the limit of an expression, say: Limit[1, c -> I, Assumptions -> Im[c] > 1] And I got a message ...
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1answer
30 views

FindRoot: equation with integral on parametric domain

Writing: ...
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1answer
40 views

How can I obtain a regularized value of this integral using Mathematica?

I want to know the regularized value of this integral. Wolfram Mathematica fails. $$\int_0^\infty \psi'(x+1)dx$$ I have two conjectures, it is either $\gamma$ or $0$. I attempted ...
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34 views

Getting an analytical form $f(x,y)$ from numerical solutions of a ContourPlot

I numerically solved a set of equations which yields a solution of the form $f(x,y)=c$. I was hoping to obtain an analytic expression for $f(x,y)=1$ (in my case $c=1$) but all I managed to do was to ...
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2answers
251 views

Find constant that allows an integral to be finite

I have a function which is a not-so complicated polynomial, depending on a parameter $c$: $$f(r,c) = \frac{-2 r^4-4 r^3-6 r^2-6 r-3}{8 r^3}-\frac{c \left(-2 r^2-2 r-1\right)}{4 r^3}$$ What I am ...
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1answer
46 views

Derivative does not correspond to function

I have an implicitly defined function, f[p], and I plot it together with its derivative. As is clear from the picture, the function is increasing, but the ...
2
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2answers
262 views

Find the area between the curves [on hold]

x = 0, x = 4, y = 2e^3x, y = e^3x + e^9 I have tried to ...
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1answer
58 views

Indefinite Integration of real-variable function gives complex logarithm [closed]

Integrating real-variable function in Mathematica Integrate[1/Sqrt[2 x - x^2], x] gives following output with complex logarithm $$ \frac{2 \sqrt{x-2} \sqrt{x} \...
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1answer
33 views

Singularity or stiff system in NDSolve

I have the following piece of code ...
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2answers
51 views

Speeding up exact evaluation of an integral with non-constant limits?

I am working on a problem that has as part of its result an integral of the form $\text{Integrate}\left[c,\left\{x_1,0,m_1\right\},\left\{x_2,x_1,m_2\right\},\text{...},\left\{x_n,x_{n-1},m_n\right\...
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3answers
225 views

How to evaluate theta function's derivative numerically?

I run into this derivative but Mathematica won't evaluate: ...
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3answers
69 views

Find residue at an essential singularity

I want to find Residue[(x^3 Exp[1/x])/(1 - x^2), {x, 0}] And when I try to evaluate this I have again: ...
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1answer
40 views

Gradient of function with arbitrary number of variables

This link answers the question on how to express the derivative of a function with an arbitrary number of variables. However, what if I wanted the gradient? More specifically, if ...
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1answer
71 views

Typing and derivating unevaluated sums

First I would like to get the output of F[x_, y_, k_] := Sum[x^(m - n)*y^m, {n, 0, k}, {m, n, k - 1}] as an unevaluated (symbolic) sum. Then, derivate F ...
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1answer
67 views

n derivatives on a function of n variables

I want to derive an expression with respect to each variable of a function $f: \mathbb R^n \rightarrow \mathbb R$. More precisely: $\left(\prod_{i=1}^n\frac{\partial}{\partial_{x_i}}\right) g(f(x_1,...
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49 views

Got MeasureDump when integrate over region with decimal parameters?

I had this problem when using Integrate over regions: ...
4
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3answers
186 views

How to evaluate derivative for implicit

If $x = \frac{1}{2}y(y^2-3)$, how do I use Mathematica to evaluate $\frac{\text{d}y}{\text{d}x}$ at the point where $y=0$? Doing f[y_] = 1/2*y*(y^2 - 3) and then <...
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1answer
77 views

Why is “1” not a valid variable? Why is there not a function plotted in the Plot function? [duplicate]

This is my code for playing this kind of stuff ...
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2answers
78 views

Definite Integration with Exponental Heaviside Step Funtions

As part of an equation of state calculator that I am working on, I have to handle a virial expansion for the equation of state. As such I have to solve for the virial coefficent, $B(T)$, which is ...
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35 views

After activating, inactive integral, output is coming same as input

I am trying to plot s w.r.t r (0,10). But because of inactive integral I am not able to. When I activate inactive integral, output is coming same as input. When I am trying to plot graph w.r.t r(0 to ...
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0answers
35 views

Implementing a generic product rule

I'd like to implement a generic rule for expanding differential forms by the product rule. For instance, I'd like to transform the following PDE: $\phi \frac{\partial \phi}{\partial t} = \phi \nabla^...
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1answer
31 views

Compare their asymptotic behaviour using the order notation [closed]

For each pair of the following functions, compare their asymptotic behaviour using the order notation. f(n) = e^n and g(n)= 3n^2
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0answers
43 views

Problem with antiderivative which involves DiracDelta

Let us execute in version 12.0 Integrate[DiracDelta[x]*DiracDelta'[x], x] DiracDelta[x]^2/2 and this lasts at least since version 7. However, the ...
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36 views

Z-Transform Solution

I am trying to solve the z-transform of the following sequence using Mathematica: x[n]=-a^n u[-n-1] The known result is $X(z)=z/(z-a)$ with ROC: $|z|<|a|$. ...
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46 views

I need a starter kit for Differential Geometry

In Modern Cosmology, the Geodesic Equation is described as:$$\frac {d^2x'^2}{dt}+\left[ \left( \{\frac{\partial x}{\partial x'}\}^{-1} \right)^l_i \frac{\partial^2 x^i}{\partial x'^j \partial x'^k} \...
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96 views

How do I solve the integral over four spherical harmonics?

I want to solve this integral ...
4
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2answers
316 views

Siginificant speed difference of Integrate between 9.0 and 12.0

I just upgraded from 9.0 to 12.0, and I found some strange difference in timings for the following expressions: a. Integrate[Exp[-a*x^8],{x,-Infinity,Infinity}] b....
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45 views

Derivative operator w.r.t a vector index and change of variables

Description and code lines: Define a scalar product to represent $A_i·A_j$ , $A_i ·B_j$ and $B_i ·B_j$, such that $A_i · A_i$ = 0. This is implemented in the following way: ...
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1answer
40 views

D: Understanding the output of 'n' th derivative of a function involving Exp[1]

I am trying to find the $n^{th$ derivate of a function involving Exp[1] as given below. ...
13
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1answer
235 views

Why the Kernel crashes on these integrals in V12?

Reported to WRI, [CASE:4331819] This is using V12, on windows 10, 64 bit. Note: these integrals work OK on 11.3 on same PC. Any idea why the Kernel now crashes on these types of integrals? ...
3
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1answer
76 views

Reduce to Piecewise

There are some similar posts on SE, but none of them seems to exactly address my issue. I need to integrate a rather complicated function (over $x>1$) separately over two domains defined as ...
0
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1answer
65 views

FindMinimum giving wrong answer [closed]

I have a very simple problem that Mathematica is giving a wrong answer to. I want to minimize the following function: $\frac{3+t^2+6\theta +3 \theta^2}{6 t^3}$ in $\theta$ and $t$. Now, plotting the ...
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1answer
61 views

How to substitute by NDSolve solution to plot a new function

Assume I got some function by NDSolve , like y(t) ...
2
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3answers
323 views

Solve command does not solve this equation!

I tried to solve the following equation with Mathematica: $\left(1-x^2\right) \left(n \left(x^4-2 x^2+5\right)-4 \pi \left(x^2-1\right)\right) \sinh (\pi x) \cosh (n x)+\sinh (n x) \left(\left(1-x^...
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0answers
63 views

How to solve this equation in Mathematica?

I wonder what is the best way to solve this equation in MA? $$ y= \frac{\partial}{\partial x} ~ \frac{\partial}{\partial x^*} f $$ where y and ...
1
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1answer
75 views

How to obtain the symbolic expression for the integral?

I need to evaluate the following expression, $\frac{1}{\Gamma(1-\alpha)} \frac{d}{d x} \int_{0}^{x}(x-t)^{-\alpha}(\exp(\lambda t)) d t, 0<\alpha<1, \lambda\in \mathbb{C}$ using Mathematica to ...
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0answers
42 views

Efficient application of the fundamental theorem of calculus

How can the following definite integral be computed with Mathematica? $$ -\frac14\int_1^{\infty}\!\!\frac{\sqrt{-4 x^4-4 x^2 z-(y^2+z)^2}}{x (x^2+z)}\theta\!\left(-4 x^4-4 x^2 z-(y^2+z)^2\right)\...
8
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130 views

The most bizarre kernel crash ever. Kernel crash in Integrate under simple different setups. Unable to find cause

Reported to WRI. CASE:4330461 V 12 on windows 10 64 bit. Note: This problem do not show up in V 11.3. Only in V12. For the last 2 hrs, I've been trying to zoom into why V12 kernel crash when ...
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0answers
33 views

Solving an integral using assumptions

I am trying to solve the integral Integrate[(Sin(theta))/sqrt((r^2) + (s^2) - 2 rs(cos(theta))), {theta, 0, pi}] with assumptions that ...
2
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1answer
58 views

Computing the definite integral of a fractional polynomial containing sin(x) and x^n

I need to compute the definite integral defined as $$\int_{-\infty }^{+\infty } \frac{\sin ({x_0}\, \omega )-\sin (x \omega )}{(\sin (x \omega )-\sin ({x_0}\, \omega ))^2+(x-{x_0})^2} \, dx\,.$$ When ...
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0answers
95 views

Complicated Integral output with Unfamiliar Regularized Hypergeometric Function

I need the solution for following integral and it has output in MATHEMATICA as: ...
4
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1answer
92 views

Real and imaginary parts of complex logarithm

I need to obtain the real and imaginary parts of the below expression with Mathematica: $(- i a - m^2) \ln(\frac{i m^2}{2 a})$, where $a$ and $m$ are real. So we have: ...
2
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2answers
43 views

Integrating a function with Max function

I'm trying to evaluate the triple integral $$\int_{0}^{1} \int_{0}^{1-x} \int_{0}^{1-\max(x,y)}1 dzdydx$$ in Mathematica. The code I'm using is simply ...
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0answers
25 views

How can I code Riccati-Bessel functions nd their derivatives?

I tried my best to code Ricatti-Bessel functions and their derivatives, but I am encountering problems. Please guide me on how to code them?