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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.

2
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1answer
13 views

A problem when integrate Cos[n*x]*Cos[k*x]

When integrate the indefinite integral Cos[nx]Cos[kx] about x, where both k and n are positive integer, the result is Pi when n equals to k and 0 when n is unequal to k. However, the code ...
0
votes
0answers
7 views

How to apply Partial differentiation w.r.t. tensors?

Let's say I have an expression like $\,a^{I}=2b^{I}+3c^{I}$ where $I$ stands for an arbitrarily large set of indices. I also know that $\,\frac{\partial a^K}{\partial c^{L}}=3\,\delta^K_L$ (equals a ...
0
votes
0answers
14 views

Solving 3 variable function

I have a function given by function1[T_,f_,d_] := Module[.......]; I defined the output to be {f,V(T,f,d)} A pair of ...
1
vote
0answers
28 views

Integrating $\frac{1}{1-x} - a\, \delta(1-x)$ $vs.$ integrating $\frac{1}{1-x}$

I find that Integrate[1/(1 - x) - a DiracDelta[1 - x], {x, 0, 1}] gives -a HeavisideTheta[0] while clearly ...
0
votes
0answers
29 views

Defining a derivative as a function does not work

I have a fairly complicated function VIcase1L[b_,\[Epsilon]_, d_] := ....... All I wanted to to was to derive it this function with respect to b and define it. ...
0
votes
0answers
32 views
0
votes
1answer
48 views

Double Integral to compute conditional expectation

Let $V=X+Y$ where $X$ and $Y$ are independent random variables, both normally distributed with $\mu=0$ and $\sigma=1$. I am interested in computing $E[X \mid V\geq c]$ where $c$ is an arbitrary ...
1
vote
1answer
49 views

Taylor's theorem approximation [on hold]

I'm struggling to determine an estimate for a function (e^-x) using the taylor theorem and getting a truncation error as well. I've tried using the series function but that doesn't let me apply a=0.
1
vote
1answer
84 views

Supposedly same integral gives different results

I need to compute the number $ \sigma_U $ for every open bounded subset of $ n $-dimensional Euclidean space defined as $$ \sigma_U=\iint_Ud(x,y)\mathrm dx\mathrm dy $$ How can I compute this? For $...
2
votes
2answers
97 views

Interpolator surface

Writing: ...
0
votes
1answer
58 views

Taking derivative of general order

How can I directly evaluate the following derivative? $$ \frac{\partial ^n}{\partial x^n}\bigg[\sum_{k=1}^{m} (m-k)! (ax+b)^k\bigg] = \sum_{k=1}^{m} (m-k)! \bigg[\frac{k!}{(k-n)!}a^n (ax+b)^{...
0
votes
1answer
55 views

Why am I getting “Erfi” for the Integral of E^x^2? [on hold]

Grandson & I have been working our way through the calculus of exponential functions. When we try to integrate E to the x^2,...
2
votes
1answer
84 views

Using DSolve to solve an integral equation

I've the following integral (or DE) that I need to solve (for $x(t)$ and all the constants are known, real and positive): $$k\cdot\theta\left(t-m\right)+(n-k)\cdot\theta\left(t-v\right)=$$ $$x(t)\...
1
vote
2answers
86 views

Validating a solution for a differential equation with DiracDelta

For the following differential equation $\displaystyle-\frac{\partial ^2\phi2 (x)}{\partial x^2}+\lambda ~[\phi2 (x)]^3-\mu ^2~\phi2 (x)=\phi2(x)~\delta(x)$ ...
1
vote
0answers
50 views

Symbolic Integral

I am in trouble with the following integral: $$rtartaruga(r)=\int \frac{\Sigma(r') }{\Delta(r') \sqrt{\Delta th}} d r'$$ Where: $$\Delta(r) =\left(a^2+r^2\right) \left(\frac{r^2}{l^2}+1\right)-2 m r$...
0
votes
0answers
30 views

Why the evaluation of this Gaussian integral takes so much time?

I want the following Gaussian function integrated for both x and y however it is not evaluated by Mathematica. ...
1
vote
0answers
9 views

Why am I getting SystemException[“MemoryAllocationFailure”] in this integral?

I'm not sure why I'm getting the error message "Throw::sysexc: Uncaught SystemException returned to top level. Can be caught with Catch[[Ellipsis], _SystemException]." when trying to compute ...
0
votes
0answers
11 views

How do i differentiate a cosx function when x is in degrees? [migrated]

i have been having a hard trouble working on excercise 4a since i cannot understand how to differentiate a trig function that has x in degrees. How is it done? .
0
votes
3answers
47 views

Drawing a diagram showing how a parabola is generated

I need help. How do I create a plot like the one shown below in Wolfram Mathematica? It's the upper-right picture, for value of p = 1/2, that I want. Is it ...
0
votes
0answers
52 views

Nested Cauchy type integration

I have a function which is in integral form: $$f_+(z)=\exp\bigg(\frac{1}{2\pi i}\oint_{C}\frac{f(\alpha)}{(z-\alpha)}\,d\alpha\bigg),$$ where $C$ is a unit circle. I want to take the Fourier ...
5
votes
2answers
135 views

Does Mathematica gives us a wrong result for the integral of a function including elliptic functions?

Calculus tells us that the differentiation of the integral of a function should be itself, but at least in one case Mathematica answers NO. I feel very confused. The new figure seems to bifurcate at ...
0
votes
2answers
115 views
0
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0answers
34 views

Numerical solution to approximate the singular integration using collocation method

I am working to solve "numerically" the following integral equation IE: u[x]=f[x]+Integrate[(1/(x-t)^(1/4))*u[t],{t,0,x}]+Integrate[(1/(x-t)^(1/4))*u[t],{t,0,1}] ...
3
votes
2answers
157 views

Eigen values of a third order linear homogenous ODE

From a system of PDEs where i used the following ansatz: $$\theta_w(x,y) = e^{-\beta_h x} f(x) e^{-\beta_c y} g(y)$$. $F(x) := \int f(x) \, \mathrm{d}x$ and $G(y) := \int g(y) \, \mathrm{d}y$ So, $$\...
4
votes
2answers
649 views

Mathematica gives an unexpected answer for Integrate [closed]

I need to integrate the following: \begin{equation}\tag{1} \frac{\sqrt{C + (1 - C) x^3}}{x}, \end{equation} where $0 < C < 1$ and $x$ is a positive variable (then $x^3 \ge 0$). When I integrate:...
0
votes
0answers
44 views

Why does Limit not compute simple limits involving integer parts?

I asked Mathematica to compute for me the following limit: Limit[IntegerPart[M a]/M, M -> Infinity] Whatever a is, the ...
0
votes
1answer
96 views

Problems with numerical integration

I tried to plot beam intensity using a function that evaluates a numerical integral, but it didn't work. Here is my code, which did not produce a result. ...
0
votes
0answers
56 views

Transform a field with derivatives

Given a function of Cartesian coordinates $(x, y)$ in $g(x, y) = \frac{d}{dx}f(x, y)-y\frac{d}{dy}f(x, y)$, I want to transform this expression to polar coordinates $(r,\phi)$. I saw a solution using ...
1
vote
0answers
46 views

Integral of the derivative

In 11.3.0 for Microsoft Windows (64-bit) (March 7, 2018) writing: Integrate[D[x*Sqrt[f[x]], x], x] I would expect: x*Sqrt[...
0
votes
0answers
30 views

How to calculate Lyapunov constants for dynamical systems using Mathematica? [duplicate]

How to Calculate Lyapunov constants for Dynamical System using Mathematics?
1
vote
1answer
131 views

Characteristic Function of 3D Gaussian with Boole

Context Given the Probability Distribution ...
0
votes
0answers
25 views

Minimum of explicit functions in vector form

For fun, related to this question about minimizing implicit functions, how can I find the minimum of an explicit one expressed as a vector? For instance, an ellipse: ...
0
votes
1answer
25 views

Simplify function with a derivative

This may be a very simple question: how can I substitute out a term involving a derivative in Mathematica? I have the following derivative: ...
1
vote
2answers
34 views

Evaluating multiple integrals

I'm trying to evaluate many long but easy multiple integrals of the form $$ \int^{\tau}_{s_n=0}\int^{s_n}_{s_{n-1}=0}\dots\int^{s_2}_{s_1=0} C(s_1,\dots,s_n)\ ds_1\dots ds_n $$ where there are ...
0
votes
1answer
58 views

Divergence and gradient operators in two dimensions

Divergence of a vector $\mathbf{a}$ can be numerically written as, \begin{eqnarray} \nabla \cdot \mathbf{a} \approx \sum_{i=0-9} w_i\mathbf{a(x+c_i)}\cdot \mathbf{c}_i \end{eqnarray} The lattice ...
1
vote
1answer
43 views

Integrating piecewise function in two variables

I want to find (and plot) the integral of a 'smooth' version of the Möbius function MoebiusMu[x]: ...
1
vote
2answers
57 views

Getting thet derivative of a user defined function in terms of an auxiliary function

I'm currently struggling with evaluation control when getting the derivative of a function (I dont know if that's the correct term here. Sorry, beginner). My function is of the form ...
0
votes
1answer
34 views

FourierTransform and Integrate don't agree [closed]

With the input f[t_] := Cos[2 Pi t] FourierTransform[f[t], t, w, FourierParameters -> {0, -2*Pi}] I get ...
0
votes
1answer
61 views

Integral of an odd function doesn't converge

I have a function $y(x)=\frac{x}{1+x^2}, x\in Reals$, it is an odd function, with no undefined points, so I expect its integral (in the range $[-\infty,+\infty]$) to be 0. But when I use Mathematica ...
0
votes
2answers
39 views

How to find value of an expression if some cases are given>

I actually need to find the value of x which have 3 cases as follows: x=0, when x<=0 x=[x], when x>0 and is non-integral ...
9
votes
2answers
430 views

On the n-th derivative of the inverse function

Given a function $f(x)$, its inverse $g(x)$ is defined as $g(f(x)) \equiv x$. In light of this, the n-th derivative of $g(x)$ can be recursively calculated as follows: ...
0
votes
1answer
55 views

Derivative of a function in undefined dimension

I have a scalar function defined on a $ n $-dimensional manifold: $ f(x_1, x_2, ..., x_n) $, where $ n $ is undefined, and $x_i$ are the coordinates. How to define something like "$∂_af∂^af$"? (I'm ...
0
votes
0answers
72 views

Potential with two variables [duplicate]

How would a procedure look like to calculate a potential$$ V(r)=-\int_0^r F(s) \,\mathrm{d} s $$ with example data: ...
-1
votes
1answer
85 views

ParametricPlot problems

I have the following code: ...
0
votes
0answers
67 views

Implicit derivative of set of equations

I have an equation and I want to take its derivative with respect to two variables. However, it contains some parameters which need to solve them together to find them (Pn, omega, gamma). These ...
2
votes
2answers
170 views

On an interesting integral

I am trying to evaluate the following integral in Mathematica $$ \int_{-\infty}^{\infty}\mathrm dk_1\int_{-\infty}^{\infty}\mathrm dk_2 \frac{m\pi}{(k_1^2+k_2^2+m^2)^{3/2}}\frac{\Delta(k_1^2-k_2^2)}{\...
0
votes
1answer
57 views

Integrate working strangely

I have the following function: f=-((2 x \[Xi]x \[Xi]z^3 (A*B) UnitBox[1/10 (-5 + x + \[Xi]x)] UnitBox[ 1/2 - z - \[Xi]z])/(\[Xi]x^2 + \[Xi]z^2)^(3/2)) where A ...
1
vote
0answers
58 views

A little magic for the Euler-Lagrange equations?

Consider a simple, one-dimensional Lagrangian: L = m (z'[t]^2)/2 - m g z[t] The built-in, partial-derivative operator D has ...
1
vote
1answer
52 views

Express result of calculation in terms of Gamma functions only

I would like to to express the result of my integration just in terms of Gamma functions. The following integral is at hand: $$ \int_0^1dz\int_0^1dy(z(1-z))^{-\epsilon}(1-y)^{1-2\epsilon}y^{-1-\...