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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1answer
23 views

Partial derivative of defined function with vector input

Background: Have written a formula for the refractive index for a glass slab (n) that depends on the variables t,a and d. Now I want to calculate for example some sort of error and I'll be needing to ...
0
votes
0answers
34 views

Nest-Recursion leads to overflow - Numerical calculation of Lyapunov exponents [on hold]

I calculate the Lyapunov exponents of several dynamical systems and want to compare the results with other algorithms. One of the most famous puplications is the work of Marco Sandri, see this paper. ...
0
votes
1answer
23 views

Boundary condition not correctly imposed for NDSolve

I notice that when plotting the below NDSolveValue ...
4
votes
3answers
152 views

About the wrong evaluation of an integral

Here is an integral I've been studying in my research and I've just realized that Mathematica $8.0$ is unable to correctly compute it. I have 2 simple questions to ask: Is my code below correct? ...
2
votes
1answer
76 views

Double integral over normal pdf gives inconsistent answer

I am using Integrate function to do the following double integral ...
2
votes
1answer
67 views

an Integral that should be doable:

Integrate[Cos[\[Beta]] Exp[I z Cos[\[Beta] - \[Alpha]]], {\[Beta], 0, 2 \[Pi]},Assumptions -> z \[Element] Reals] correct answer is [ I think ]: ...
10
votes
4answers
355 views

Correct way to integrate a certain function

I want to integrate $$ \int_{0}^{2\pi} dt_1 \frac{a + b*\cos(t_1 - t_2)}{c + d*\cos(t_1 - t_2)} $$ where $a, b, c, d, t_2$ are real numbers and $c + d > 0$ & $0 \leq t_2 \leq 2\pi$. I used ...
7
votes
6answers
994 views

Trying to prove that $x\sin(\frac{\pi}{x})\ge\pi \cos(\frac{\pi}{x})$ for $x\ge 1$

Consider the function f[x_] := x Sin[Pi/x] I want to prove that this function is increasing for $x\ge 1$. This can be done with the first derivative. We have to ...
26
votes
1answer
321 views

many indefinite integrals do not evaluate in 10.1, looking for the cause

Many integrals no longer evaluate in V 10.1 when they did in 10.0.2 Here are some 23 integrals as an example, that all produced results in V 10.0.2, but now all returns unevaluated! I am hoping ...
2
votes
0answers
57 views

Strange Integrate messages / $RecursionLimit being ignored

Introduced in 10.1.0 The following behavior was observed in Mathematica 10.1 (Windows 64 bit) When attempting to evaluate the following integral, Mathematica outputs several messages which appear ...
0
votes
0answers
43 views

NIntegrate returning exactly what I inputted [closed]

I just want to get an approximate number for this definite integral. Help would be appreciated :)
4
votes
1answer
123 views
0
votes
1answer
62 views

Can't obtain a closed form expression for an integral

I am trying to find the result of quite a complicated double integral. While the first part (being a Principal value integral) gave no particular problems, the second one is not providing any closed ...
10
votes
4answers
3k views

How can I differentiate Numerically?

Mathematica has two ways to integrate: Integrate and NIntegrate. But what about D? ...
-6
votes
1answer
37 views

Need help with Integrating user defined function [closed]

Can I integrate the following? z,a,L,k are variables integration is with respect to r ...
5
votes
1answer
143 views

Integration over a convex combination of a region: $\int_{\Omega} (w_1 z_1 + w_2 z_2)^{1-\sigma} d (z_1, z_2)$ where $\Omega = \{ z_1 + z_2 = 1\}$

Take $w,z\in R^{n}$. I am interested in integrating (as generically if possible) $$\int_{\Omega}(w \cdot z)^{1-\sigma} d z$$ Where the domain of $\Omega$ is $1$ dimension, and includes the convex ...
4
votes
1answer
1k views

Lie-Bracket of two vector fields

I'm new to Mathematica (installed couple of hours ago) and I need to compute a few Lie brackets between two vector fields $f$ and $g$. $$ f\left(\mathbf{x}\right) = \left( \begin{array}{c} ...
0
votes
1answer
46 views

Adapt the code to produce a three column table containing the values of n, etc. (Numerical integration)

Here is a “procedural” program that we wrote in my class, implementing the rectangle rule of numerical integration: ...
0
votes
0answers
39 views

Using mathematica to to find region of integration of triple integral [closed]

essentially what i want to do is see the region of integration so i can figure out the limits i need to integrate over i am able to plot each bound individually but can put them together to make the ...
0
votes
1answer
84 views

Plotting multiple functions in 3D plot [closed]

I would like to plot the flowing into one 3D region. So I know the region of integration the functions are $z=1-x-2y, z=0,x=y$ I want to have all graphed on $1$ plane so I can see the region of ...
0
votes
0answers
43 views

Calculus with Mathematica, books, interesting examples [duplicate]

I'm looking for a book where I could find informations and interesting examples regarding examine functions of one variable like continuity of functions, limits, differentiability in Mathematica. I'd ...
0
votes
0answers
33 views

Computing integral involving derivatives

I want to evaluate an expression $\int_C g(\nabla f)ds$. This is an integration along a curve $C$ in a plane given as a function of some parameter $s$: ...
-1
votes
0answers
26 views

Adapt the code to produce a three column table (rectangle rule of numerical integration) [duplicate]

I was given the following program implementing the rectangle rule of numerical integration: ...
7
votes
0answers
45 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 ...
2
votes
3answers
137 views

How can I get the exact value minimum of this function?

I want to find min of the function $$\frac{1}{\sqrt{2 x^2+\left(3+\sqrt{3}\right) x+3}}+\frac{1}{\sqrt{2 x^2+\left(3-\sqrt{3}\right) x+3}}+\sqrt{\frac{1}{3} \left(2 x^2+2 ...
0
votes
1answer
71 views

Four integrals, how can I expedite the calculations?

I have to calculate the following multiple integrals (4 integrals) ...
9
votes
2answers
228 views

Mathematica integration failure - new or old?

Bug introduced in 7.0.1 or earlier, persists through 10.1 Consider the following integral: Integrate[Log[a Cos[x]^2 + b Sin[x]^2], {x, 0, 2Pi}] This takes a ...
23
votes
3answers
622 views

Symbolic integration error

fixed in 10.1 (windows) I'm running Mathematica 10.0.0 and encountered a disturbing error in the symbolic integration of a rather simple function ...
7
votes
3answers
302 views

Negative integral of a positive function

fixed in 10.1 (windows) For a parameter $t\in (0,1)$ $Assumptions = t ∈ Reals && t > 0 && t < 1 I define an obviously positive function ...
24
votes
2answers
307 views

Bug in ArcLength?

fixed in 10.1 (windows) With Mathematica 10.0.2: ArcLength[Line[{{0, 0}, {1, 0}, {2, 0}}]] ArcLength[Line[{{0}, {1}, {2}}]] (* 2 *) (* 2 *) However, ...
8
votes
3answers
281 views

Definite integral incorrectly giving a nonreal value

fixed in 10.1 (windows) In Mathematica 10.0, when I enter N[Integrate[Sqrt[1+x^3],{x,-1,3}]] , I get a nonreal value. (I.e., the imaginary part is nonzero.) ...
4
votes
1answer
451 views

Integrating over Bessel Function erroreous? (Hankel Transform)

Bug introduced in 8.0.4 or earlier and persists through 10.0.2. The Hankel Transform is given by Integrate[f[x] x BesselJ[0,x t],{x,0,Infinity}] It is ...
0
votes
1answer
176 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 ...
5
votes
2answers
351 views

How can I get the inverse of my function?

The original function is $$f=\frac{1}{e^{\frac{\phi }{k t}}-1}-\frac{m+1}{e^{\frac{(m+1) \phi }{k t}}-1}$$ I want to express $e^{\frac{\phi }{k t}}$ about $m$ and $f$, so I tried: ...
0
votes
0answers
29 views

numerical integration with parameter

Now I'm trying to integrate a function numerically. But it shows always an error message. x=y is not a valid limit of integration I know that this function ...
0
votes
1answer
49 views

PDE of real-world system, integral boundary condition

I've stripped all the physical-significance for clarity, but I know that u[x,t] will be everywhere positive and continuous. here are the equations in Mathematica code: ...
0
votes
0answers
77 views

How can I verify double integral solution?

I have a double integral as follows \begin{equation} 1-(\int_0^\infty (\frac{Cje\cdot Cte \cdot e^{\frac{-w}{Cte}}}{(Cte+Cje \cdot w)^2} + \frac{e^{\frac{-w}{Cte}}}{(Cte+Cje \cdot w)}) ...
0
votes
0answers
28 views

How to solve this linear equation? [migrated]

$(\frac{dy}{dx})^2+6\frac{dy}{dx}+4y=x^{2}e^{2x}$ how to solve this. this is linear equation. what type of equation is it? first order?
4
votes
2answers
118 views

Well-defined symbolic integral leading to ConditionalExpression

I would like to determine a closed-form expression for the following symbolic integral $$ \int_{-1/2}^{1/2} \!\!\!\! \mathrm{d} x \int_{-1/2}^{1/2} \!\!\!\! \mathrm{d} y \, \frac{1 + b x + c y}{1 + e ...
1
vote
1answer
81 views

How can I simplify a triple integral with exponentials?

I want to simplify the following triple integral with exponential terms. \begin{equation} \int_0^\infty\int_0^\infty \int_0^\infty \frac{1}{R\,G} e^{-p(1+R\,a)-q\, b \frac{1+R\, a}{1+G\, x}}\, ...
4
votes
1answer
204 views

Lagrange Multiplier

We are asked to maximize and minimize $f(x,y)=4xy$, given the constraint $4x^2+y^2=8$, using the Lagrange Multiplier method. First, I enter the functions f and g. ...
3
votes
3answers
56 views

Multiple integrals where the number of integrals is aribtrary

I know that I can do, say, a triple integral such as $$\int_{a_x}^{b_x}\int_{a_y}^{b_y}\int_{a_z}^{b_z}f(x,y,z)\,dz\,dy\,dx$$ with the input ...
10
votes
1answer
377 views

Symbolic Integration along contour: branch cut problem?

Context Following this question on path integrals in the complex plane, having defined again a numerical and symbolic integrator along a path as ...
14
votes
3answers
2k views

How can I implement the method of Lagrange multipliers to find constrained extrema?

I want to form the function $h=f-\lambda_{1}g_{1}-\lambda_{2}g_{2}$ where $f$ is the function to optimize subject to the constraints $g_{1}=0$ and $g_{2}=0$ so that I can solve the first partial ...
-4
votes
1answer
41 views

Show steps to compute limit [closed]

I'm trying to use Mathematica to compute a limit, because I have no idea how to compute it by myself. The limit should be: $$\lim _{n\rightarrow \infty }\dfrac {2^{n+1}-n-2}{2^{n}}$$ Any ideas?
0
votes
1answer
47 views

Cauchy Principal Value integral- no result is obtained [closed]

I have a particular Cauchy Principal Value integral that I need to numerically solve for my thesis research. It is the following $$ ...
4
votes
2answers
240 views

Using Mathematica to confirm Bernoulli's inequality

I have several challenges that I want to confirm is true. I have chosen this one because it is rather simple (proof by induction). There are times when I do not want to spend ages trying find proofs. ...
0
votes
0answers
26 views

Use implicit differentiation to find the slope of the tangent line to the curve 4 xy^3+ 5 xy= 45 at the point ( 5 , 1 ) [duplicate]

Use implicit differentiation to find the slope of the tangent line to the curve 4 xy^3+ 5 xy= 45 at the point ( 5 , 1 )