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.
5,048
questions
2
votes
2
answers
618
views
Symbolic linear algebra gradients/matrix calculus
Can Mathematica generate symbolic expressions for gradients?
For example, if $x_1$ and $x_2$ are two points, could I get Mathematica to generate expressions similar to the following?
$\frac{\partial ...
- 123
0
votes
0
answers
80
views
Doing vector calculus with element-wise defined formulas and arbitrary-sized vectors [duplicate]
Currently I am trying to make Mathematica handle vector expressions defined element-wise. For example,
But I can find no way to do this. Every way I tried requires a fixed-sized vector and returns a ...
- 143
4
votes
2
answers
170
views
Symbolic Integrate fails after almost 1 hr calculation
Mathematica 9. I want this
Integrate[(
Cos[t ω]^2 Cos[α + t ω])/(-1 +
A Cos[α + t ω])^2, {t, 0, (2 π)/ω}]
Can Mathematica handle with it?
I tied different ...
- 101
4
votes
2
answers
175
views
How to define action of derivative (D) on a user-defined operator f
I have a symbolic function (which is in fact an operator) f[expr, x] where the second argument is a local variable for f (kind ...
- 19.7k
2
votes
1
answer
408
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Residue failed to reproduce this residue in a paper
This problem is a more clearly state of my original problem which is off topic, so I reconsider my problem carefully based on previous discussions, and gives the following valid code to reproduce my ...
- 1,323
0
votes
1
answer
252
views
Integral of $(\sin x)^3(\cos x)^3$ [closed]
For the integral of (sin x)^3(cos x)^3 I got 1/6(cos x)^6 - 1/4(cos x)^4 + C. This seemed to work but my book had 1/4(sin x)^4 - 1/6(sin x)^6 + C. This is similar to what I got but instead of cosine ...
- 1
2
votes
0
answers
147
views
Why does Expectation function compute, while explicit integration does not?
The code:
Expectation[1 + 1/y, {y \[Distributed] ExponentialDistribution[1]}]
outputs
1 - EulerGamma
While,
...
- 21
2
votes
3
answers
394
views
NDSolve and differentiation of Abs
I have found several questions about the derivative of Abs and how it is not defined in the complex plane. What I have not found yet is a precise and simple ...
- 839
1
vote
1
answer
242
views
Problem in limit of exponential function
I want to obtain
Exp[-α/(1 - α)] /. {α -> 1}
The output is Indeterminate. But I calculate without Mathematica and I ...
- 423
1
vote
1
answer
398
views
Need help on simplifying the integral of a square root of a polynomial
I have the following expression
a = 4 Cos[af / 2]^2; b = 1 - 5 Cos[af / 2]^2; c = Cos[af / 2]^2;
s = Sqrt[a*x^4 + b*x^2 + c]
and I need the integration of $s$ in ...
- 725
4
votes
1
answer
377
views
Using Vector Operations In Mathematica
I would like to calculate
$$ \mathbf T = (\mathbf B \bullet \nabla) \mathbf B$$
Where nabla (the upside down triangle) is the grad operator $(\partial/\partial x,\partial/\partial y,\partial/\...
- 4,366
1
vote
1
answer
779
views
Evaluate an integral with parameters
I already asked this on stack overflow but was redirected here.
I am trying to solve the (highly non-linear) following system of equations :
with
and $g_0$, $\Lambda$ are positive constants which i ...
- 111
4
votes
4
answers
651
views
Integrating SquareWave - don't understand result
[Acknowledged as a bug persisting in V12.2]
Integrate[SquareWave[t], t] // InputForm
Gives as output
...
- 16.8k
3
votes
2
answers
804
views
Integrate on a contour in the complex plane
I want to integrate the function
f[z]= z + Conjugate[z]
over a circle of radius 2 centered at the origin.
For the sake of stating something that I have tried:
<...
- 1,661
2
votes
0
answers
295
views
Solving Differential Equation for derivative
I am new to mathematica and am trying to solve a differential equation. Actually, I am not entirely sure if the system can be called differential equation.
I am interested in finding out the second ...
- 33
3
votes
2
answers
299
views
Application of # for the $n^{th}$ derivative
Let's say that I have
x^2+x
Is there a way to map $x$ to the first derivative of a function and $x^2$ to the second derivative of the same function? According to ...
- 992
0
votes
2
answers
227
views
How to set a line coordinate for a symbolic line integral over a curve? [closed]
I have the following closed curve, defined by a set of points.
...
- 2,192
1
vote
0
answers
297
views
How to do this integral of Hypergeometric functions in Mathematica?
I've tried to integrate by part, but it seems that Mathematica is still not able to integrate.
...
- 455
8
votes
2
answers
376
views
Calculating sum of BesselJ[n, x]
My friend has a sum in his research paper that looks like this
$$
\sum_{n=-\infty}^{\infty}\frac{J_n^2(x)}{n-\kappa}.
$$
He was able to calculate this sum analytically, by substituting the ...
- 215
5
votes
2
answers
1k
views
Double integral over the region enclosed by several lines
The question is:
Calculate the double integral $$\iint_D (18x^2+51xy+30y^2)^2\,dx\,dy$$
where $D$ is an area enclosed by the following lines:
$$6x+5y=9,\quad 6x+5y=-9,\quad 3x+6y=7, \quad 3x+...
- 53
3
votes
1
answer
1k
views
Plotting the 3D region that determines a volume
I need to plot this with Mathematica:
Use a triple integral to determine the volume of the region below $z = 4 - x\,y$ and above the region in the xy-plane defined by $0 \leq x \leq2,\ 0 \leq y \...
8
votes
1
answer
367
views
Symbolic and numeric limit disagreeing on branch cut of ArcTan
Bug introduced after 8.0.4 and before 9.0.1 and fixed in 11.1.0.
Investigating comments to my previous question lead me to the following problem:
...
- 341
9
votes
2
answers
599
views
Integrate and NIntegrate disagreeing over branch cut of Sqrt?
Bug introduced after 8.0.4 and before 9.0.1, and fixed in 11.1.0.
In Mathematica 10.2 I'm trying to integrate this piecewise continuous function, but Integrate and ...
- 341
1
vote
1
answer
249
views
nonatomic expression expected when trying to sum of the integrals of two functions
newbie to MMA.
After computing the following two integrals:
...
- 23
0
votes
2
answers
65
views
Issue while trying to compute the integral for the coefficients of a triangular wave
I was trying to compute these integrals in order to check if I was correct:
or in code
...
- 103
2
votes
0
answers
178
views
What does Mod^(1,0) mean in mathematica output [closed]
I am trying to take he derivative of mod(remainder).
In:= D[Mod[x, y], x]
Out = Mod^(1,0)[x,y]
I cannot understand the meaning of Mod^(1,0). What does the ...
- 121
1
vote
2
answers
1k
views
Integral over real valued function becomes complex
I've tried to solve the following integral, but I get a complex solution even tough it should only have a real part.
$$f(x)=- \frac{10^{-20} x}{0.99005- e^{10^{-12}x}}$$
Now I want to calculate the ...
- 119
1
vote
1
answer
396
views
How to integrate with integer parameter in Mathematica?
I would like to integrate the following expression with N being Integer
...
- 155
10
votes
1
answer
450
views
Wrong limit: Limit[(1 + (-1)^n/n)^n, n -> Infinity]=1 (Mathematica 10.4 and W-Alpha)
Bug introduced in year 2002 (?) and fixed in 11.0
Since $(1-\frac{1}{n})^n\to 1/e$ and $(1+1/n)^n\to e$, the sequence $(1+\frac{(-1)^n}{n})^n$ has no limit as $n\to\infty$, but has limits for odds ...
- 274
11
votes
1
answer
413
views
Vastly incorrect answers obtained by increasing WorkingPrecision with modified Bessel functions
Bug introduced in 7.0 or earlier and fixed in 11.0
This is a follow-up to this question regarding numerical instabilities occurring with modified Bessel functions. In trying to explore J.M.'s answer ...
- 743
2
votes
0
answers
427
views
Tricky inverse Laplace transform
I'm trying to compute the inverse Laplace transform of $f(s) = s^c/(N + s^{ir} )$ where $c,N \in \mathbb{C}$ and $r \in \mathbb{R}^+$ using the Bromwich integral
$$
F(t) = \frac{1}{2 \pi i} \int_{- ...
0
votes
1
answer
159
views
Definite and indefinite integral give different results
I'm trying to find an expression that, given $x$, $y$, and $R$, gives the indefinite 2D integral of a function at the circle centered on $(x_0, y_0)$ with radius $R$.
With $g$ as defined below, I'd ...
- 103
2
votes
1
answer
228
views
Why does NDSolve and NIntegrate not give the same result? [closed]
I have plotted solution of two equivalent equations one in Integral form (right chart) the other in Differential form (left chart) using NDSolve and NIntegrate but they give me completely different ...
- 181
6
votes
2
answers
387
views
Strange phenomenon occurring in analytic integration result involving Bessel functions
For the following integral,
Integrate[x^2 Exp[-a x^2 - b x^4], {x, -∞, ∞}, Assumptions -> {a > 0, b > 0}]
Mathematica gives the following analytic ...
- 743
1
vote
1
answer
219
views
How can the same Mathematica version give different symbolic results on different machines?
I'm testing the following code:
...
- 7,152
1
vote
1
answer
405
views
Replace higher order derivatives given lower order derivative
I have an unknown function f[x1,x2] whose partial with respect to x1 is identically zero. I would like to make sure that ...
- 336
3
votes
1
answer
5k
views
Defining functions as derivatives
I'm working on a model that requires a sum over a function that uses various coefficients. In order to make the sum easier to perform, I am trying to keep things general through the use of a ...
- 33
4
votes
2
answers
249
views
Problems with integrating over a region
Following the example of this answer:
Find the volume of the region defined by $|x|+|y|+|z|<4$
I tried to compute the following integral over a region by using ImplicitRegion. However, when I ...
- 43
12
votes
1
answer
211
views
Something wrong when I apply Derivative on a SparseArray
Bug introduced in v5.0 or earlier, and persisting through v11.0.1 or later
There, I create a SparseArray using command
...
- 183
13
votes
1
answer
2k
views
How does a Pringle lose its curvature?
Nom!
As part of a bigger project, I've was writing some code to calculate the scalar curvature of surfaces of the form $z = f(x,y)$. This uses a general calculation of the scalar curvature to produce ...
- 828
0
votes
1
answer
146
views
Tedious integral
I'm new to Mathematica so I'm not familiar with its potential use to solve tedious symbolically integrals. I'm trying to solve the following one:
...
- 103
1
vote
1
answer
226
views
integral involving square of exponential integral
I'm trying to compute the integral
$$
\int_{a}^{\infty}\frac{e^x}{x}[\mathrm{Ei}(-x)]^2\,dx,
$$
where $\mathrm{Ei}$ is the exponential integral, and $a>0$. The obvious
...
- 755
1
vote
0
answers
25
views
2
votes
1
answer
250
views
From which version Mathematica will be available Symbolic n-th derivative? [duplicate]
On www.wolframalfa.com Symbolic n-th derivative works.
Edited:
On new version Mathematica 11.1 symbolic n-th derivative works.
- 13.9k
0
votes
1
answer
2k
views
Assumptions with Multiple Rules
What is the correct way to add more than one assumption while integrating?
I am attempting to evaluate the following integral and Mathematica is just stalling, so I'm wondering if my assumptions are ...
- 1,050
3
votes
1
answer
139
views
Dog chases his tail ! - "parametric differential/Integral equation"..?
I have the following situation where I am interested in the function $m(t)$
$$ \frac{dm}{dt}=4T(t)^{3}+T(t)^{2} $$
$$ T(\tau)=T_{0}-\int_{0}^{\tau}(\frac{dm}{dt})dt*Q_{S} $$
Is there a way to solve ...
- 33
2
votes
1
answer
526
views
How to linearize an expression involving derivatives?
I want to linearize the expression involving function $y(x,t)$
$$[y^3(y_x-y_{xxx})]_x$$
by assuming $y(x,t)=y_0(t)+y\prime(x,t)$, where $y\prime(x,t)$ is a small variation. In other words, I will ...
- 1,229
7
votes
3
answers
827
views
Integrate to calculate enclosed area
I am trying to 'use an integral in polar coordinates to calculate the area enclosed by this curve':
The curve is: $r=\sin 2\theta$, $\theta \in [0, \pi]$ which I believe is already in polar form.
I ...
- 71
1
vote
2
answers
424
views
How to compute this integral?
I'm trying to calculate this integral :
$$I(z,k,a)= \displaystyle\int_{1}^{\infty} t^2\, \text{ArcTanh} \left(\sqrt{\frac{t^2-1}{t^2}} \dfrac{k}{z}\right)\, e^{-a\,t} \, dt$$
Where :
$\textrm{...
- 415
3
votes
2
answers
3k
views
How to plot a volume for triple integral enclosed by surfaces
My main issue (multivariate calculus), I want to plot the body made within the surfaces
y + z = 4, y = 4 - x^2, ...
- 33