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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1
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0answers
22 views

Are the conditions in ConditionalExpression necessary or sufficient?

I computed a limit and obtained it in the form of a ConditionalExpression (see below). Substituting the variables with numerical values in the resulting formula ...
2
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0answers
43 views

Step by step solution of a limit of a sequence - Calling Wolfram|Alpha in Mathematica

I'd like to visualiza the step-by-step solution of the following limit by calling Wolfram|Alpha in Mathematica, but I don't know how to specify that n is a positive integer variable: ...
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0answers
32 views

Plotting a complex function under transformation

I am trying to plot the unit circle $\vert z\vert=1$ under the transformation $z\to z+\frac1z$ using mathematica. In general I want to plot a complex function $f(z)$ under the transformation $\phi(z)$....
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2answers
75 views

Primitive of a continuous function over $\Bbb R$

A continuous function over $\Bbb R$ has a primitive that is also continuous over $\Bbb R$. However, it often happens with Mathematica and other CAS that the result is not continuous, especially with ...
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0answers
50 views

Small issue with PDE initial condition

I'm trying to solve the Fokker-Planck equation $$\frac{\partial p}{\partial L}(L, \eta)= \frac{\partial}{\partial \eta}\left[\left(\eta^{2}-1\right) \frac{\partial p}{\partial \eta}(L, \eta)\right] = ...
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1answer
98 views

«Hidden rocks» of the algebra of indexed values

Some time ago, our community generally believed that one should avoid operating indexed variables such as Subscript[a,1]. I remember to have several times adviced ...
2
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0answers
54 views

Laplace Inverse [closed]

Mathematica did not give any out put when i am going to compute Laplace Inverse for a little bet complected function, Kindly if you have any comment or you advice any code to compute the inverse here....
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0answers
53 views

Speed up Integrate around Max/Min or Piecewise

Integrate around a convex combination of two functions containing Min or Max or ...
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0answers
35 views

smallest distance to place the capital? [migrated]

A country has four cities where should the capital be built. it is desirable that the sum of distances of the cities from the capital be as small as possible? how do i solve it ?
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0answers
20 views

Numerical evaluation of derivatives of pair

I have a set of ordered pairs. How can I take calculus action on that like derivative or integration? $$ N(E) = \{(2, 3), \cdots \} : N=2 , E=3\\ \mathrm dN(E)/\mathrm dE = ? $$
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0answers
88 views

A bug Integrating Piecewise functions

I have found a very puzzling bug (version 12.0), where integrating the sum of a piecewise function with a DiracDelta with a variable integration limit causes errors,...
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0answers
58 views

Teach Mathematica analytical continuation of the gamma function

If I ask Mathematica to compute the gamma function for me Integrate[Exp[-s] s^(a - 1), {s, 0, Infinity}] It dutifully returns to me ...
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2answers
31 views

Integrating a defined Function of 2 variables

I have defined a function p(V,T): p[V_, T_] = (R*T/(V - b)) - (a/(V^2)) And then I used: Integrate[p[V, T], {V, V1, V2}] ...
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48 views

gradient of a numerical function

Is there a way other than interpolation to evaluate the numerical differentiation using Mathematica? I have a function A(x,y) and I want to obtain the gradient of this function. ...
1
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0answers
38 views

Nesting differential operators

Suppose I define differential operators $d=x\frac{\partial}{\partial x}$ and $L=g(x)d^4$ (so that $L(f)=g(x)(x\frac{\partial}{\partial x})^4f$ ) and I want to expand the differential operator $L=a_0+...
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1answer
66 views

Efficient way to evaluate numerically product sines functions

I'm trying to perform integration of the following matrix. ...
10
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3answers
566 views

Finding 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 structure of Black Hole, OUP). In page 98 of the ...
0
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1answer
33 views

How to define a function through so that it has an integral of a fixed value?

I have a quite complex function of two-variables, let's say $\alpha(a,b)$ that has a parameter $c$ and I want to set this $c$ so that $\int_{-\infty}^{\infty} \alpha(a,b) \: \mathrm{d}a \: \mathrm{d}b ...
5
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1answer
122 views

Wolfram Language 12 says this absolutely convergent series does not converge. Is there any similar example?

I am reading "Lectures on complex function theory" by Takaaki Nomura. In this book, there is the following example: $\sum_{n=1}^{\infty} \sin(\pi(2+\sqrt{3})^n)$ converges absolutely. But ...
3
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1answer
86 views

Display returned integral containing assumptions with integral sign

From solving the PDE example of the general 1 dim heat equation in Nasser Abbasi's site https://www.12000.org/my_notes/pde_in_CAS/maple_2019_and_mma_12/insu157.htm#x182-1810003.1.1 (his problem No 151)...
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0answers
60 views

Tell MMA that $\int_0^1 \alpha \beta(x) dx= \alpha \int_0^1 \beta(x) dx$

Why is it that MMA excludes alpha from the integral in the first case but not in the second? ...
3
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1answer
52 views

Evaluate $\int_{0}^1 \frac{\sin\|a v\|}{\|a v\|} a \mathrm{d}a$ as $\frac{1-\cos(\|v\|)}{\|v\|^2}$

Is it possible to have Mathematica evaluate the following integral? $$I=\int_{0}^1 \dfrac{\sin\|a v\|}{\|a v\|} a \mathrm{d}a$$ where $v$ is a vector. By hand: since $a\in[0,1]$ and with $\mu=\|a ...
4
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3answers
215 views

How to simplify this formula?

$$ \int \frac{1}{\sqrt{1-2 x^3}} \, dx $$ Integrate[1/Sqrt[1 - 2 x^3], x]// FullSimplify The result is very complex. Then I want to differentiate it with the ...
5
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0answers
127 views

Changing the integration limits takes a loooooooooooooooooog time to evaluate

Integrate[x*Sin[x]/(1 + Cos[x]^2), {x, 0, π}] π^2/4 But ...
1
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1answer
45 views

Can this PDE be solved through DSolve?

I was trying to solve the linear advection pde, using DSolve. I was able to do it using NDSolve, but I was curious to see if Mathematica was able to give the well-...
6
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0answers
230 views

How to find that limit by mathematica? [migrated]

Limit[Sum[2(2k)^(1/(2k))-k^(1/k),{k,n+1,2n}]-n, n -> ∞] to solve by hand, $$\sqrt[y]y=e^{\frac{\ln y}{y}}\sim1+\frac{\ln y}{y}$$ $$2\sqrt[2k]{2k}-\sqrt[k]k\...
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0answers
38 views

Finding sequence limits from matrix determinants

I have a problem whereby I want to calculate the limit of a sequence of determinants of matrices of increasing dimensions using the DiscreteLimit command. To illustrate this, I use an example with a ...
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1answer
151 views

How does MMA use $\varepsilon-\delta$ language to find the limit?

In the help document of Reduce function, I can find an example of using the definition of limit to find the asymptote: ...
0
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0answers
74 views

Integration of multidimensional Gaussian

How could the following integration of a 6-dimensional Gaussian be achieved? Are there some techniques? (MMA code below) $$\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty ...
1
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1answer
67 views

How can I plot the “sol” in the following?

I'm trying to plot the following differential equation for different values of P but no succes: ...
4
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3answers
417 views

Why does Mathematica refuse to evaluate my integral?

Here is the integral for which I want a symbolic result: Integrate[x^(z - 1)PolyLog[2, x]/(1 + x), {x, 0, 1}] But the output is the same as the input without any ...
0
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0answers
26 views

Finding a Laurent polynomial that behaves at a pole like a given function [migrated]

I need to find a polynomial $P$ such that $P(1/x)$ behaves like given function $f(x)$ at zero (their difference vanishes), even if $f(x)$ has a pole at $0$ or unknown order. P.S. I need to find a ...
4
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2answers
150 views

Fast way to the Taylor series expansion coefficients of multivariable function?

Is there a fast method to get the coefficients of Taylor series expansion of function $f(x_1,x_2,...,x_d)$ with maximal summed partial derivative up to $n$, where $d,n$ can be relatively large (for ...
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1answer
67 views

Complicated integration [closed]

How to solve the following integration g1 with f[z] and g[z] as constant wrt integration- ...
1
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1answer
55 views

Limit in Infinite

My code: ...
6
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3answers
202 views

Integral $\int_{d_1}^{d_2} \int_{-L/2}^{L/2} \int_{-L/2}^{L/2} \frac{1}{(x^2+y^2+z^2)^3} dx dy dz$

I'm trying to calculate the following integral in Mathematica, but it seems it doesn't return an analytical closed form, neither when I give numeric values for both $d_{1,2}$ and $L$. $$∫_{d_1}^{...
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0answers
48 views

Find vectors describing two interconnected pyramids in Euclidean space

I need the help of competent linear algebra specialists in Mathematica. In the Euclidean space, two tetrahedrons are turned at different angles. Their vertices coincide with the beginning of the base ...
0
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1answer
34 views

How to discritize a function domain (break xy-plane to a grid/mesh of squares with vertex values for evaluation)? [closed]

I wrote a code using MATLAB to discritize a function's 2D domain into squares/rectangles, whose vertex data (x,y) I can use to evaluate the function at. Then I sum these values over all vertices to ...
1
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0answers
87 views

How can I get this integral to Compute

I'm trying to compute the following expression and get an analytical answer but Mathematica ends up spitting back the input expression. Hoping someone can shine some light on this. ...
1
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1answer
77 views

Integrate 2D Gaussian over a segment

I have the following 2D Gaussian: ...
2
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1answer
124 views

Constructing and plotting a special periodic function

I need to program and plot the following special function in Mathematica 12.0: $$ a_{1}(t)= \sum_{n\geq 1} \beta_{n}(t), $$ such that, for every $ n \geq 1 $ $$ \beta_{n}(t)= \displaystyle \sum_{...
1
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1answer
35 views

define function as a result of gradient returns unexpected result

This is probably very simple problem, but what am i doing wrong? c[x_, y_] := x*y; Grad[c[x, y], {x, y}] d[x_, y_] := Grad[c[x, y], {x, y}]; d[1, 1] Returns: <...
4
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1answer
54 views

Area of circle segment

Why does it take so much time computing a simple area of segment of a circle radius $a,$ width $h<a?$ Integrate[Sqrt[x ( 2 a - x)], {x, 0, h}] Would it ...
4
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2answers
157 views

How to calculate volume under $ f(x, y) $?

I have the following function: σ = 0.3; μ = 5; cy =0; cx=0; f[x_,y_]=Exp[-(1/2)((Sqrt[(x-cx)^2+(y-cy)^2]-μ)/σ)^2]/(σ*Sqrt[2*π]) Plotting this with: ...
2
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1answer
54 views

WolframAlpha does not give stepwise solution for a simple twodimensional integral

Why does the behave this particular way: ...
0
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0answers
50 views

How to simplify an expression with implicit vector functions and cross product operation?

1.The pattern is set in order to calculate the integral of a derivate of a function (to return original function, first fundamental theorem of calculus) ...
0
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0answers
37 views

Radial Mathieu functions, divergence problem

I am working on a project that requires the utilization of even Mathieu functions. This is the definition of my functions: Even Mathieu function: ...
2
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2answers
370 views

Integration of probability density function

I would like to integrate the function so that I know the probability between values between 0 and 90000 ...
1
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1answer
95 views

Is it possible to use an assumption that a parameter is an odd or even number?

I am trying to integrate: Integrate[(Exp[x]*(x^m)/((Exp[x] + 1)^2)), {x, -Infinity, Infinity}] I know that for odd m the result is zero. How can I tell ...
0
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0answers
27 views

Inconsistency in complex line integral using 2 parameters vs 1 parameter

I am trying to numerically verify that, over a closed circular path, the line integral of a physics-inspired complex quantity $z_Q = P \cdot \frac{\partial}{\partial Q} S$ is the same regardless of ...

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