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
53 views

Convergence of numerical integration with parameter in the region of integration

I would like to solve the integral: $ f(a)=\iiint\limits_{\mathcal{R}(a)}\frac{\sqrt{12\left(\left(\frac{x-y}{x+y}\right)^2+\left(\frac{y-z}{y+z}\right)^2+\left(\frac{x-z}{x+z}\right)^2\right)}}{\exp(...
10
votes
1answer
339 views

Limit of sequence of functions behaving strange

I'm trying to determine the limit of the sequence of functions $$f_n(x)=\left(\frac{1}{\pi}\arctan(n x) + 1/2\right)^n. $$ I define ...
14
votes
1answer
223 views

Is it a bug in Limit?

Bug introduced in 9.0 and fixed in 10.0.0 Mathematica 9 on Windows and WolframAlpha says Limit[Sin[Sqrt[x+1]]-Sin[Sqrt[x]],x->Infinity] is Interval[{-2,2}]....
6
votes
3answers
382 views

Using Grad, Div, etc in polar coordinates

I'm trying to use the newish functions Grad and Div to work with a PDE with circular or spherical symmetry and getting what looks to me like a wrong result. However, I'm not sure of this--perhaps I ...
2
votes
3answers
6k views

Step-by-step definite integration

I am interested in obtaining the intermediate steps for rather straightforward symbolic definite integrations (e.g., from an elementary calculus text). As an example, for $\int_0^1 x\,dx$ I'd like ...
1
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1answer
559 views

Why is my expression not a quantified system of equations? [closed]

Who can help me to understand why my code is wrong? It is giving me a message saying "is not a quantified system of equations and inequalities"?
0
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1answer
155 views

How to integrate a function of a direction, over the hemisphere

The Mathematica documentation Integrate over Regions gives an example of how to simply integrate over a sphere (surface): ...
1
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1answer
95 views

Help with singular integral

I'm new in Mathematica. I need solve a singular integral like $$\int_a^b \frac{A_0 + A_1 s + A_2 s^2 + A_3 s^3}{(B_0 + B_2 s^2 + B_4 s^4)(t-s)}ds$$ where $A's$ and $B's$ are reals and $0<a<t&...
1
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1answer
92 views

How can I avoid a scoping problem when differentiating?

I simplify my real problem as follows: I define a rule function as Clear[rule]; rule[i_] := tt -> i; the real rule function is not this simple. I define a ...
1
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0answers
51 views

Combining Manipulate and NDEigenvalues [closed]

I think there may be a quick solution to this problem which I can't seem to fix at the moment. The question is how do we eliminate the error in the following ...
33
votes
1answer
743 views

How to augment the realm of functions Mathematica thinks it knows how to integrate symbolically

My question involves extending the functionality of Integrate over specific integrals in the most generic manner. Specifically, is it possible to "hack into" <...
2
votes
3answers
179 views

Calculate and plot the tangent hyperplane to a function of three variables

If f:= x^3 + y*x*z - z^2 + x*z and if {{1,2,1},{2,7,8},{6,0,2},{9,1,1}} are points, how can I find the tangent plane to <...
29
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2answers
2k views

Bug in mathematica analytic integration?

I found Mathematica provides me a wrong answer for a relatively simple analytically solvable integral: ...
9
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1answer
751 views

Symbolic scalar-by-matrix derivative

Let's say I want to calculate the following scalar-by-matrix derivative $$\frac{\partial}{\partial A} \text{tr} \left[(\vec X^T A)^T (\vec X^T A)\right],$$ with $\vec X$ and $A$ being a $n \times 1$ ...
28
votes
1answer
450 views

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

Bug introduced in 10.1 and fixed in 10.2 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 ...
10
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1answer
150 views

How to predict the degree of the first series coefficient?

Given an expression f that is a function of x and a number x0, what is the least integer <...
6
votes
3answers
363 views

How to do algebra on unevaluated integrals?

I am working with functions calculated from a set of general basis functions. ...
13
votes
1answer
554 views

Why is this infinite series wrongly computed by Mathematica?

Bug introduced in 7.0 and fixed in 10.0 Could you let me know if Mathematica (newer versions) is able to correctly compute this one? ...
1
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1answer
119 views

Strange behavior of `FourierSinCoefficient`

FourierSinCoefficient works normally like this and the result is checked with Integrate. ...
2
votes
1answer
155 views

Function to calculate finite difference

I wrote a function to calculate multivariate finite difference of different orders ...
4
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1answer
143 views

How to collect terms with z-derivative?

my equations are very long (several pages). Here I will provide simple example: ...
5
votes
1answer
163 views

How to do matrix calculus in Mathematica?

I would like to differentiate Ax with respect to A where A is a matrix. Now, obviously, I know the derivative is x but for more nasty expressions, is there a way to differentiate expressions with ...
1
vote
1answer
58 views

Evaluate list of determinant of Hessian at list of points

If f:= x^2y + z^3 + xz; and if {{1,2,9},{2,4,6},{9,5,9},{3,3,1},{6,5,3}} is a list of points. I need to evaluate the hessian at ...
6
votes
1answer
120 views

Unexpected Behavior from Integrate with PrincipalValue -> True

Bug introduced in 9.0 or earlier and persisting through 10.3 or later The following code, taken from my answer to 97024, ...
3
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1answer
165 views

Multivariable Calculus Chain Rule

I am getting ready to write a notebook for the chain rule in Multivariable Calculus for my students. I do know: ...
0
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1answer
50 views

if-conditional tracing a half-sphere [closed]

I have a 2D discrete plane (x,y) and I would like to assign an amplitude to each point in the plane. Each dimension runs from -1 to 1. Say I want to assign an <...
4
votes
2answers
278 views

Convergence and value of a complex power series

I've done a little math and I got the following power series expansion of $\log z$ about $z_0=-2+i$. $$\log z=\log(-2+i)+\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n(-2+i)^n}[z-(-2+i)]^n$$ I've shown that ...
4
votes
1answer
173 views

Bad performance of Integrate (and WolframAlpha) for an Integral of Bessel function of the first kind

The following returns unevaluated in WolframAlpha. Also in my machine Mathematica needs quite a lot of time to compute it. ...
4
votes
0answers
119 views

Puzzled by Assumptions [duplicate]

I don't know if this has already been discussed. Integrate[BesselJ[2 m + 1, x], {x, 0, ∞}, Assumptions -> m ϵ Integers] ...
20
votes
2answers
3k views

Usage of Assuming for Integration

For some reason, when I enter the following integration in Mathematica Assuming[{k ∈ Integers}, Integrate[ Exp[ I k t], {t, -π, π}]] the result turns out to be 0....
12
votes
6answers
486 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 a ...
1
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2answers
172 views

Find the explicit solution of the integral

I want to calculate the integral: assuming that we are familiar with the relation: The answer should be: The code that I've used is the following: ...
0
votes
2answers
126 views

Why does Integrate[] return an expression with 'True' in it?

I am running Mathematica 10 on Windows 10. I'm trying to analytically evaluate a definite, principal value integral: ...
7
votes
2answers
422 views

Check homework integration in Mathematica

Given in my homework I have to compute (by hand) $$\iint\limits_{x^2+y^2\leq 1}(x^2+y^2)\,\mathrm dx\,\,\mathrm dy.$$ My solution so far: Let $f(x,y)=x^2+y^2$ and $K=\{(x,y)\in\mathbb{R}^2:x^2+y^2\leq ...
1
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1answer
90 views

Find Values of a and b for a Limit

I am having trouble finding the values of a and b for which $\lim \limits_{x \to 0}$ $(\frac{\sin (ax) + bx)} {(x^3)}) = -32/3 $. In Mathematica I have entered: ...
7
votes
1answer
644 views

Integrating over Bessel Function erroreous? (Hankel Transform)

Bug introduced in 8.0.4 or earlier and persists through 10.2. The Hankel Transform is given by Integrate[f[x] x BesselJ[0, x t], {x, 0, Infinity}] It is self-...
2
votes
1answer
156 views

Derivatives of vectors

I have an issue with derivatives of vectors ...
0
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1answer
47 views

How can evaluate this function [duplicate]

Suppose this function F[c_, n_, s_] := Module[{}, Abs[s - 1 + (-1 + c)^n]] I calculate its second derivative on c ...
1
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1answer
76 views
2
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2answers
93 views

Strange failure of Series and Derivative

I just spend three hours and posted two Questions trying to figure something out, and it turned out all the confusion was caused by this mysterious quirk. I want to expand g[x,v] in v at v=0, using ...
0
votes
1answer
102 views

Hessian of function at many points

If f:= x^3 + 2yz + 5 x^2z is a function and if {2,0,1},{3,9,2},{8,5,6},{1,1,7} are points. I would like a Hessian of ...
0
votes
1answer
47 views
4
votes
0answers
112 views

Integral of DiracDelta giving an unusual answer

I have been getting a number of seemingly inconsistent solutions to integrals of Dirac delta functions in which the integrand evaluates to DiracDelta[0] at one of ...
1
vote
2answers
206 views

How would you calculate a contour integral? [duplicate]

How would you be able to calculate a contour integral of 1/(((z-1)^2)*(z-i)) over the contour |z-1| = 1? Not sure how to type ...
4
votes
1answer
130 views

A 1D numerical integral Mathematica cannot compute, from physics

A well know result in theoretical physics is that a sum over Matsubara fermionic frequencies, i.e.: $$ S = \sum_{n=-\infty}^{\infty} h(\omega_n) \hspace{32pt} \omega_n=(2n+1)\frac{\pi}{\beta} $$ can ...
1
vote
0answers
135 views

Solving a functional equation

Suppose I have two functions f[x,y,z] and g[x,y,z], both analytically defined. I want to solve for a function ...
5
votes
2answers
182 views

Finding the equation of a level curve

Is there a way in Mathematica to determine a parametric equation of a level curve in Mathematica. For example, consider: ...
0
votes
1answer
79 views

Finding the limit of a complicated expression [closed]

I've never used Mathematica before, so I'm really struggling to use it right now. I have a complicated function $\qquad \chi_{N}=\frac{[(2\cosh(J))^{N+1}(\lambda_{0}+\lambda_{1})^{N}((\mu_{0}+\mu_{1}...