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.
3,141
questions
0
votes
1answer
46 views
Find analytic solution for integral only defined for even integers
I would like to calculate the integral $$\int_0^{2\pi} dx \sin^6\left(\frac{x}{2}\right) F\left(\frac{4-n}{2}, \frac{4+n}{2}, \frac{1}{2}, \cos^2 \frac{x}{2} \right)^2$$
where $F$ is the ...
2
votes
2answers
47 views
Double integration with a piecewise function gives wrong result
I am attempting to solve a double integral
$$I(t) = \int_0^t dt' \int_0^{t'} dt'' f(t'')$$
of a piecewise function of the type
$$f(t) = \begin{cases}1 & \text{for } 0 \leq t < a \\ -1 & \...
0
votes
0answers
21 views
Multiply Taylor expanded matrices with multiple variables and different order
I want to multiply rotation matrices (Euler matrices) that are expanded in Taylor series to a different order. When I use Series on the separate matrix, the multi-...
0
votes
0answers
39 views
Chain rule applied to a kinematic problem
I have a very simple question.
How can I enter and solve the following equation in Mathematica?
thank you for your help
dv/dt = ds/dt dv/ds
A supplementary question on the same subject. What is the ...
1
vote
1answer
50 views
Plot Point On StreamPlot
How would I go about plotting a single point in the $x,y$ plane given by;
$$\left(\frac{\lambda d -b\mu}{ad-bc}, \frac{a \mu -c\lambda}{ad-bc} \right)$$
Where we can take $a,b,c,d,\mu$ to be positive ...
1
vote
0answers
45 views
Solving PDE IVP with inequalities in domains
I want to solve the following problem in Mathematica:
I used the following Mathematica code. I am unsure whether this is the correct way of entering the assumptions for the initial and boundary ...
0
votes
0answers
37 views
Plot inside Streamplot [duplicate]
I am looking to include plots in the $x,y$ plane that represent the lines that the fixed points of a dynamical system follow as the variable $l$ is varied. In doing this, I am trying to represent ...
0
votes
2answers
80 views
0
votes
0answers
9 views
The convergence/divergence of lim x>0 ln(1+f(x))/f(x) when f(x)=sin(1/x)x^2 [migrated]
This question came up between me and my teacher when we discussed whether the following
when lim x>a f(x) = o
does lim x>a ln(1+f(x))/f(x) = 1
is true or not.
It is widely known that
lim x>0 ln(1+...
3
votes
3answers
139 views
On correct use of Limit for multivariable function [closed]
V 12.1 on windows.
This limit $\lim_{(x \rightarrow 0,y\rightarrow 0)} \frac{x^2-y^2}{x^2+y^2}$ depends on the direction. So the limit does not exist, or could be written as Maple does it, which is $...
0
votes
1answer
48 views
0
votes
0answers
48 views
Is zero multiplied by infinity indeterminate or zero?
Is $0\times\infty$ indeterminate or zero? Mathematica says the result is zero.
Limit[0*n, n -> \[Infinity] ] or ...
4
votes
1answer
78 views
Can you force Integrate[] to find a complete symbolic solution for all variables?
(As I've asked on the Math StackExchange and on a related previous question), I am interested in getting a complete symbolic solution to the integral of an expression with a lot of unassigned ...
3
votes
1answer
87 views
Results from Integrate not matching NIntegrate
I am interested in getting a completely-symbolic form of an integral of an expression. But I noticed that when I compare my symbolic integral with a numerically calculated integral, I see that there ...
6
votes
1answer
186 views
Changing the integration limits takes a long time to evaluate
Integrate[x*Sin[x]/(1 + Cos[x]^2), {x, 0, Ļ}]
Ļ^2/4
But
...
9
votes
2answers
146 views
Finding simplifying substitutions for an integral involving limits and integrand
[The following is based on a William Lowell Putnam Mathematical Competition problem.]
Consider the definite integral:
$I = \int\limits_2^4 \frac{\sqrt{\log (9-x)}}{\sqrt{\log (9-x)}+\sqrt{\log (x+3)}...
0
votes
0answers
63 views
Convective derivative in cylindrical coordinate
My question is similar to this one about convective derivative. But how do I adapt it to the cylindrical coordinate? Simply changing Grad to cylindrical coordinate ...
10
votes
2answers
763 views
The time-like geodesics (orbits) in the Schwarzschild spacetime
I am trying to plot Schwarzschild's orbit without invoking the geodesic equation.
As a reference I am using Chandrasekhar's Book (The Mathematical theory of Black Holes, Oxford University Press). In ...
1
vote
2answers
54 views
0
votes
1answer
103 views
Why can't Mathematica integrate this integral?
I'm trying to estimate this integrate:
Here Ī is the Heaviside theta function. And I can use
PR[k_] := 2.1*10^-9 (k/0.05)^.04
I want to solve this integral for ...
0
votes
1answer
73 views
Unclear result from Leibniz Integral Rule
I was trying to answer this question: $\frac{\mathrm d}{\mathrm dt}\int_{t-\mathrm d1}^t \int_h^t f(s) \,\mathrm ds\,\mathrm dh$. I got the accepted answer by hand (using the Leibniz integral rule) ...
2
votes
1answer
62 views
Mathematica integrates centered functions, but can not integrate shifted ones
Mathematica seems to integrate this function:
$\int \limits_{-\infty}^{\infty} d w\, \frac{\sin ^2\left(\frac{1}{2} wt \right)}{w^2}
\frac{\frac{\gamma ^2}{4}}{ \left(w^2+\frac{\gamma ^2}{4}\right)}$,
...
0
votes
0answers
35 views
Derivative for conditional expectation of non-independent gaussian variables is increasing
Consider two non-independent gaussian random variables:
$$(t,c)ā¼\text{BiNormal}[(šš”,šš),(šš”,šš),š]$$
I'm interested in understanding when (i.e. for which values of the distribution ...
0
votes
1answer
63 views
0
votes
1answer
43 views
Understand output of Integration
Could someone help me understand the output of the following Integration in Mathematica?
Specifically,
(a) what is the meaning of "True"?
(b) Why is the answer in two parts which seem to overlap ...
2
votes
1answer
65 views
Define function of n variables and integrate [closed]
Im looking at n particles and functions of their coordinates which I want to integrate over:
...
0
votes
1answer
40 views
Derive arbitrary steps between two expressions
Can someone show me the syntax to find the algebraic transformations between:
$\sqrt{1+(\frac{dy}{dx})^2}dx$
and
$\sqrt{dx^2+dy^2}$
And more generally, how to get the algebraic derivation between ...
0
votes
0answers
90 views
Rubi/Mathematica gives different values for the same integrand
Consider this integral:
$$
\int_0^\pi d\theta \sin\theta\, {1 \over \gamma^q}\left((1-\gamma^2)^2\sqrt{1 \over 1-\left(1-(1-\gamma^2)^4\right)\cos^2\theta} - 1\right)^q
$$
for $0\leq \gamma < 1$. ...
1
vote
1answer
68 views
complex number from the integration of a real integrand
Consider this integral:
$$
\int_0^\pi d\theta \sin\theta\, {1 \over \gamma^q}\left((1-\gamma^2)^2\sqrt{1 \over 1-\left(1-(1-\gamma^2)^4\right)\cos^2\theta} - 1\right)^q
$$
for $0\leq \gamma < 1$. ...
1
vote
1answer
58 views
Two different Integration results for the same integrand in Mathematica
I want to calculate
$$
\int_0^\pi d\theta\sin\theta\, \left(\sqrt{1 \over 1-\left(1-(1-\gamma^2)^4\right)\cos^2\theta} - 1\right)^q
$$.
For the simplest case of $q=1$ we have
...
1
vote
1answer
114 views
Integration not working
Can't seem to get this integration to work. s, n, b, b, l are all constants.
...
0
votes
0answers
47 views
Can anyone help me to get symbolic expression for the following code?
Using Mathematica I did some manipulation on the function $\sum _{s=-m}^{m} e^i \lambda ^{-(2-d) s} \left(1-e^{i y \lambda ^s}\right)$.
And got the following output:
(1) when I took $\text{m}$ in ...
1
vote
0answers
62 views
Faster Way of Plotting a Multivariate Integral with No Closed Form and Maximized
I have a double integral that looks like this.
$$J=\int_0^2\int_{-2}^2 \sin(y\cdot B_z+z\cdot B_y+\phi_y+\phi_0)\,dy\,dz$$
$\phi_y$ I am defining as different functions of y, e.g. $\phi_y=0$, $\...
4
votes
1answer
86 views
Jacobian Matrix?
I am doing
r[psi_, th_, phi_] := {Cos[(phi + psi)/2]*Cos[th/2],Cos[(phi - psi)/2]*Sin[th/2], Sin[(phi - psi)/2]*Sin[th/2], Sin[(phi + psi)/2]*Cos[th/2]}
and ...
2
votes
1answer
107 views
Line integration given tangent vector
I have the following problem: I know a unit vector in spherical polar coordinates, let's call it $\vec{e}_{s}$, in the form $\vec{e}_{s} = f(r,\theta)\vec{e}_{r} + g(r,\theta)\vec{e}_{\theta} + c\vec{...
1
vote
1answer
31 views
How to get the output for the following code involving differentiation and integration?
I need to evaluate the following Mathematica code to get a numerical value as my output. But during compilation I get the error message
NumericalMath`FixedPrecisionEvaluate::precbd: Requested ...
2
votes
4answers
261 views
Series expression of a RootSum object
Mathematica was not able to calculate the definite integral of a trigonometric function of mine:
...
1
vote
1answer
64 views
Fractional Euler-Lagrange equation [closed]
Using Mathematica, I need to perform the derivation of the fractional Euler-Lagrange equation as derived in this article.
Regarding this, I searched many links, but I was not able to master the ...
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7,1741 gold badge13 silver badges35 bronze badges
3
votes
3answers
211 views
How to calculate arbitrary area on surface of sphere?
I am trying to calculate the solid angle subtended by arbitrary-shaped loops on a sphere's surface.
First, I parametrize circular loops by:
$$\theta(t,k_{x0},r) = k_{x_0} + r \cos(t);$$
$$\phi(t,k_{...
4
votes
2answers
341 views
Why doesn't Integrate evaluate an elliptic integral?
My code is
Integrate[ 1/Sqrt[(x - a) (x - b) (x - c) (x - d)], {x, a, ā},
Assumptions -> 0 < d < c < b < a]
I know this can be ...
1
vote
2answers
159 views
Why is my integral not being evaluated?
I am trying to integrate a function Gamma, but the output returns the input. I have the following code:
...
0
votes
1answer
68 views
Error in Taylor's Series using Mathematica
How many terms should be used in taylor series expansion of the function f(z) = e^z around z = 0 for a specific value of z = 30 + 30 i to get an error of less than 0.05 using Mathematica?
...
5
votes
3answers
385 views
Calculating curvature of a contour
I have an equation of a scalar field in the form
$$f(x, y) = x^2 + y^2 + xy + c$$
I want to find the curvature of the contour of the curve at $f_c = f(0.5, 0.5)$.
So I need to calculate the ...
2
votes
0answers
50 views
Why Integrate does not obtain this conditional result automatically for orthogonality of cosines?
V 12.1 on windows.
I remember asking something similar many years ago. I was hoping Mathematica now could have done this automatically:
$$
\int_{-\pi}^{\pi} \cos (n x) \cos (m x) \, dx
$$
For ...
1
vote
0answers
63 views
Why doesn't Mathematica simplify? [closed]
Consider the following:
Simplify[Integrate[x^(a), {x, x1, x2}], x1 > 0 && x2 > 0 && a > 0]
Mathematica doesn't simplify this for me. I ...
13
votes
1answer
4k views
Kramers-Kronig relations
I am trying to calculate the change of the refractive index from the change of the absorption coefficient using the Kramers-Kronig relations, in Mathematica.
...
2
votes
1answer
52 views
Plot the expression using NIntegrate for the limiting case
I need to plot the below equations:
$W^{+}(x,\tau,\alpha)]=-\frac{2m}{\pi\alpha}\exp[{\alpha x}]\times\int_{0}^{\infty}\frac{G(\omega,\tau_c)}{1+\omega^2}\cos(2\tau_c\omega\sqrt{1+\omega^2}-2\tan^{-1}...
2
votes
2answers
228 views
On evaluating an integral related to Chern numbers
I am trying to evaluate the following integral in Mathematica
$$
ā«_{-\infty}^{\infty}\mathrm dk_1ā«_{-\infty}^{\infty}\mathrm dk_2 \frac{mĻ}{(k_1^2+k_2^2+m^2)^{3/2}}\frac{Ī(k_1^2-k_2^2)}{\sqrt{[Ī(k_1^...
5
votes
2answers
730 views
Bug in Integrate?
In v. 11.1.0.0 Mathematica for Linux
Integrate[Sin[Cosh[t]], {t, 0, Infinity}]
returns
Integrate::idiv: Integral of Sin[Cosh[t]] does not converge on {0,[...
6
votes
0answers
6k views
What's the most difficult multidimensional integral that Mathematica has solved?
I am interested in benchmarking numerical integration methods and am trying to develop a wider set of difficult multivariate examples. For my particular methods, I only want to look at non-negative-...
