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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$ \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)}}}...
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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: ...
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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, ...
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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: ...
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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): ...
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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.
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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:$...
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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: ...
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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 ...
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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 (...
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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 \...
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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 ...
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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: ...
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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]...
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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?
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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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: ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\bar{}$ ...
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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 ...
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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;\...
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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}\...
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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} \...
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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) ...
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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 ...
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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{\...
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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 ...
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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
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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
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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 ...
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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 ...
4
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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: ...
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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, ...
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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
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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, ...
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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 ...
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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: ...
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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 ∞?
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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 ...
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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: ...