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.
2,733
questions
0
votes
1answer
52 views
$ \lim_{a\rightarrow0^{+}}\frac{\intop_{0}^{1}e^{\frac{-30}{\sqrt[15]{a^{16}*\log\left(\frac{1}{x}\right)}}}*xdx}{e^{\frac{-32}{a}}}<+\infty $
I am trying to verify the following inequality using Mathematica, but without any success.
$$
\lim_{a\rightarrow0^{+}}\frac{\intop_{0}^{1}e^{\frac{-30}{\sqrt[15]{a^{16}*\log\left(\frac{1}{x}\right)}}}...
1
vote
1answer
80 views
Why is my integral not being evaluated?
I am trying to integrate a function Gamma, but the output returns the input. I have the following code:
...
3
votes
2answers
86 views
Why am I getting a negative value from $\int_{0}^{\infty}x^{\frac{-9}{8}}*e^{-16*x^{\frac{-1}{8}}-\frac{2*x}{0.5^{9}}}dx$?
In Mathematica 9, I want to evaluate the following integral:
$\qquad \int_{0}^{\infty}x^{\frac{-9}{8}}*e^{-16*x^{\frac{-1}{8}}-\frac{2*x}{0.5^{9}}}dx$
The result should be positive. However, ...
3
votes
3answers
128 views
A discontinuity in an integral
I have a function which I'm trying to integrate, which is perfectly smooth and Mathematica sees it as well defined everywhere in the region over which I'm integrating it. The function is:
...
1
vote
1answer
103 views
Solving Ginzburg–Landau Equation in MMA 11.3
Based on the PDE model proposed by @Schumacher Solving a second order coupled PDE system, the one dimensional Ginzburg–Landau Problem I would like to solve such benchmark test:
namely:
u(x): ...
1
vote
1answer
71 views
Writing a function that can perform n-times fractional Caputo derivative of order alpha to a function u(x) inductively
I wrote this code, but it fails to give the intended answer which is D^(1/2) three times of x^2.
any suggestions to correct this code or write a new one.
0
votes
1answer
35 views
DSolve not satisfying such kind of initial conditions
I am trying to solve this equation in MMA 11.3
$2 \epsilon s^{\prime \prime}+\frac{1-s}{2 \epsilon}=0$;
BCs: $s^{\prime}( \pm 1)=0$ and $s\left(0\right)=0$;
The analytical solution is expressed as:$...
12
votes
1answer
987 views
Laplacian and DiracDelta
It is known that:
$\nabla^2 \dfrac{1}{|r-r'|} = - 4 \pi \delta^3(\vec{r}-\vec{r}')$.
If I do that with Mathematica, I find:
...
4
votes
2answers
117 views
Solve a logarithmic equation by NSolve
I'm trying to solve:
$$1 - |x| Log(\frac{|1+x|}{|x|}) = 0.84$$
by using NSolve:
NSolve[1 - Abs[x] Log[Abs[(1 + x)/x]] == 0.84, x]
Mathematica doesn't return ...
4
votes
1answer
276 views
Implementing positivity constraints over a six-dimensional hypercube
This question involves the same subject matter as my previous one (How can one achieve the most accurate estimates of certain six-dimensional integrals under specific constraints?), but with another (...
0
votes
0answers
23 views
Define the partial derivatives of a multivariable function
I have some vector $K = [K_1, K_2,...K_n]$ which is a function of variables $(x,y)$ i.e. $K = K(x,y)$
I want to symbolically calculate the partial derivatives e.g. $\partial_x K$, $\partial_x \...
0
votes
0answers
105 views
Need help with symbolic integration
I have
f[x_] := g[x]/g'[x]
FullSimplify[Integrate[f[x], x]]
I'm trying to get F[x] in terms of ...
4
votes
2answers
145 views
To plot branch cut of logarithm
I like to see the branch cut of the function:
$$1 - z \ln[(1+z)/z].$$
If I plot it in the complex plane:
...
5
votes
1answer
112 views
Visualizing Riemann surface (two branches) of logarithm
I'm trying to plot two branches of the complex multi-valued function: (In a previous post - linked above-, Mathematica found the branch cut of the following function between -1 and 0)
$1-z\ln[(1+z)/z]...
0
votes
0answers
36 views
Normalized eigenvectors are different depending on when the limit is calculated
I have this code:
...
9
votes
5answers
2k views
Is there a way to rewrite integrals in Mathematica using u substitution?
In Mathematica, can I give it an integral and a few substitution rules and have it rewrite the integral in terms of those variables?
0
votes
0answers
55 views
Unexpected difference between integral and summation?
I am trying to integrating something like: Integrate[Exp[-i*(k*x+k*z)]*Exp[-(x^2+z^2)],{x,-largenumber,largenumber},{z,-largenumber,largenumber}]
My issue is that ...
0
votes
1answer
77 views
Is it possible to use mathematica to find the derivative of a function with respect to some “ax” where “a” is a constant?
Mathematica evaluates the derivative of most complicated functions within a fraction of second. But while doing analysis sometimes we might be in a situation to differentiate or integrate a function ...
0
votes
2answers
44 views
problem with constant variables
Variable x is supposed to be constant, but the program calculates for each variable x, 100 numbers of x. I want all my x variables to have the same definition, but the program constantly changes each ...
3
votes
0answers
58 views
Derivative of eigenvalues
I work with 4x4 Hermitian matrices (r). I want to calculate a derivative of a function f[t,r] (ff[t_,r_]=1/2*D[f[t,r],t]), where the function f depends on the absolute value of the eigenvalues of r. ...
9
votes
3answers
960 views
How can I solve for the intersection points of two ellipses?
I need to find the points at which the two ellipses implicitly defined by
$\qquad y^2=4-4\,x^2 \quad[E1]$
and
$\qquad (1-(x/2))^2+(y-1)^2=1 \quad[E2]$
intersect.
So I isolated $y$ in E2 and ...
0
votes
0answers
34 views
How to obtain the gradient of a function wrt a vector?
I'm new to mathematica and I wanted to obtain the gradient of the following function:
where b_k(s) is the cubic B-spline basis function associated with control point
k.
I tried this code:
...
1
vote
1answer
708 views
Integration of linked PDFs over probability simplex
I'm trying to integrate the following expression:
f[a1,p1+p2+p3+p4]*f[a2,p2+p3+p4]*f[a3,p3+p4]*f[a4,p4]
Where
...
1
vote
1answer
79 views
How is the best way to integrate using loops
I'm using Mathematica, and I want to integrate a function f[wr] in wr using some method that works with a Table/Do/For in the variable wr.
I tried to use something like the Riemann's sum, to evaluate ...
0
votes
0answers
39 views
How to solve this DE to get a matrix?
I have this DE
$ g^{m \bar{n}} g^{r\bar{p}} \partial_t {g}_{m r} \partial_t {g}_{\bar{n}\bar{p}} = X_{i\bar{j}}~ \partial_t{f}^i (t) ~ \partial_t{f}^\bar{j} (t) ~~~ ~~~~~~~~~~~~(*) $
Where ...
3
votes
1answer
51 views
Implementing Simpson's rule with two variables
For one of my lab project I need to integrate a given function (in this example, f1, using Simpson's ruls. When only one variable ...
0
votes
0answers
43 views
Discrete inverse Laplace transform
I'm trying to find a method for obtaining the inverse Laplace transform of a series of discrete data points. I have tried deriving an analytical function from the data and then using that function ...
0
votes
0answers
33 views
Plotting matrices
I have this expression:
$\int g^{m \bar{n}} g^{r\bar{p}} \partial_t {g}_{m r} \partial_t {g}_{\bar{n}\bar{p}} ~ d^6 y = f(t)$
Where m,n,p,r=1,2,3 and the $\bar{}$ ...
1
vote
1answer
89 views
Replacing one of the integrals in the multiple integral with an analytical solution
Problem consist in simplifying complicated multiple numerical integral, where I know analytical solution for a subset of them. Integrand is of the form
...
1
vote
1answer
51 views
Evaluate a nested Gaussian integral
I would like to evaluate the following integral. I suspect it just doesn't have an analytic form but thought I'd check to see if there's anything I'm missing,
$\int_{-\infty}^{\infty} \mathcal{N}(z;\...
1
vote
2answers
94 views
Derivative for arbitrary number of times
I want to take the derivative of a function with respect to $ t $ for an arbitrary $ k $ number of times at the point $ t = 0 $ using Mathematica.
The function is
$$
f(t) = \frac{\mathrm e^{kt}\...
0
votes
0answers
72 views
Integrating an expression containing the error function, perf [migrated]
I am trying to integrate a formula, but I can’t figure out how to do it. Its integrand involves a special function — erf, the error function.
Here's the formula:
$$
f(a, b, c) = \int_{0}^{+\pi} \...
0
votes
0answers
46 views
What is the representation of the Harmonic Number being used by Mma in this result?
The Fourier Transform of the function
F[x_] = (m/Sqrt[\[Lambda]])*Tanh[(Sqrt[x^2]*m)/Sqrt[2]]
where all variables are real, and $m>0$ is given by (Mma 11.0)
...
0
votes
1answer
50 views
defining rule for derivative on arbitrary function
Is there a way in Mathematica for defining rules for derivatives on undefined functions? For example, I would like to define a rule such that for any function $f(x,y)$, when I take the $x$-derivative ...
1
vote
1answer
28 views
Mixed partial derivatives with respect to n variables
I have the following generic set of variables:
x = {x1, x2, ..., xn}
nu = {0, 1, 2, ..., n - 1}
order(nu) := Total @ nu;
I would like to calculate:
$$ \frac{\...
3
votes
2answers
121 views
Plotting the inverse function of a complicated function
So I have a function
F[x_] = Assuming[{Element[x, Reals], -1 < x < 1},
Integrate[1/Sqrt[(x^2 - 1)^2 + alpha*x], x]]
I'm now interested in the ...
5
votes
2answers
70 views
How can I help SumConvergence give the right result?
I've been trying to use the SumConvergence on the following series:
SumConvergence[1/(n Log[n] Log[n Log[n]]), n]
This ...
2
votes
1answer
48 views
Implementing the Peano-Baker Series with Mathematica
I would like to calculate the Peano-Baker series with mathematica, namely an integral expansion to approximate the state transition matrix:
Say I have a matrix $A(t)$ and I want to calculate:
$\Phi(...
3
votes
1answer
95 views
Reconstructing a function from its gradients
I have a list of the components of the gradients, $\partial f/\partial x_i$, of a function $f(x_1,x_2,\cdots)$. Is there some neat way to reconstruct the function $f$?
One approach to doing this ...
0
votes
0answers
56 views
Solving a quadruple integral of view factor
I'm solving an analytical quadruple integral in Mathematica, but the app can't solve it. I don't know if it is because that I do a bad formulation of the integral or maybe, Mathematica is not able to ...
2
votes
1answer
43 views
4
votes
1answer
89 views
Line-Dirichlet Boundary NDSolve
I have defined such a Dirichlet boundary conditions which fixes the nodes in a line, however, the results show that only two nodes are fixed, not the nodes in a line,
Code:
...
0
votes
0answers
39 views
Analytically integrating bivariate Gaussian PDF
I have this somewhat complicated expression in terms of the following 8 variables (note: actually, $X$ and $Y$ are variables, and $a,b,c,d,e,f$ are longer expressions as a funtion two other variables, ...
0
votes
0answers
22 views
Unitary group as a region
I would like to compute the area of some subspaces of $SU(n)^d$ defined by group equation (basically $U_{i_1}^{e_1}\dots U_{i_k}^{e_k}=1$) . I know that there is the function ...
6
votes
3answers
108 views
Symbolic integration of logmultinormal distribution
I have been trying to integrate and compute moments of LogMultinormalDistributionin Mathematica, but the integrals don't seem to evaluate. Say for the first moment, ...
0
votes
0answers
50 views
How to use replace for only one term in a long expression in Mathematica?
I am doing some Cauchy integration and in some intermediate steps I have something that looks like that:
(..many_terms...+I*\epsilon(k1-p1))/k1
where ...
1
vote
1answer
62 views
Finding the limit of this neat solution
In an earlier post, I asked "Can Mathematica find an expression for the distribution of the median of N i.i.d. random variables?". JimB found the very neat solution:
...
3
votes
1answer
72 views
Evaluating an integral combining a Bessel function with some other functions [closed]
How can I evaluate the integral of $j_1 ^2(x)\exp(-bx)/x$ from 0 to ∞?
0
votes
1answer
87 views
Performing a contour integration in Mathematica for a contour starting at $1$ and ending at $-\infty$ while avoiding the origin?
I would like to compute the following integral $I(k)$ in Mathematica to check if the result equals something I 'feel' is correct but I have no experience with contour integrals in Mathematica.
I want ...
0
votes
0answers
52 views
Mathematica gives uncessary condition for integration of $(b - s)^\alpha(s - a)^\beta$
When I compute the integral of $(b - s)^\alpha(s - a)^\beta$ on $[a,b]$ by the following command:
...
