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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1answer
339 views

How to code around known MMa special-case failures?

Fourier Analysis and Signal Processing often require these integrals on functions f[x] that won't be known in advance. ...
7
votes
2answers
2k views

How to change coordinates of integration?

Is there some built-in routine, some easier method to change variables of integration, without/before solving the integral? Say I have ...
7
votes
2answers
281 views

How to simplify the integration result of the following program?

In the following program, the fec[x] is a function depending on $x$ while q3[t] and q5[t] ...
7
votes
1answer
1k views

How to find the smallest root

I have a continuous, differentiable, monotonic, bounded function called F[t]. If t -> Infinity then ...
7
votes
2answers
822 views

Getting an InterpolatingFunction from a ContourPlot

I have a function, say minimizeme[Ω_][β_][ϵ_] = ϵ^2 Ω - Log[2 (Cosh[2 β] + Cosh[2 β ϵ])]/(2 β); I want to find its critical points in $\epsilon$ for a given ...
7
votes
1answer
129 views

Old fashioned region method?

Just stumbled across this idea in http://users.rowan.edu/~hassen/Mathematica/Volume%20III/Chapter%2015.pdf. ...
7
votes
1answer
472 views

Mathematica 9 can't integrate this function but earlier versions could

Integrate[ ArcTan[x]/(1 + x) Log[(1 + x^2)/2], {x, -1, 1}] I used Mathematica 9.0.1 on Windows7 32bit, Mathematica 9 cannot compute this, but Mathematica 8 gives ...
7
votes
1answer
293 views

Series expansion for small ratio of variables

I have a messy expression of variables that I would like to simplify under the assumption that certain ratios of the variables are small. For example, consider $\sqrt{x+y}\;$ expanded for small ...
7
votes
2answers
800 views

Integration with vector coefficients

I asked this same question in Mathematics, and it was suggested I might try here. I'm more comfortable with Maple, but if I can get Mathematica to do what I'm after, so much the better. Basically ...
7
votes
1answer
92 views

How to calculate residues using different branches of the logarithm

My goal is to be able to compute the residues of something like $z^{\alpha} R(z)$ where $0 < \alpha < 1$, $R(z)$ is rational, and the exponential can be chosen with any branch. I found a way to ...
7
votes
2answers
320 views

Integrating a function over a surface integral

From a first principles bandstructure calculation I get an energy scalar field in three dimensions $E(x,y,z)$. It's now easy to plot a constant energy (contour)-surface for dedicated values ...
7
votes
2answers
776 views

Mathematica 10 cannot solve definite integral [duplicate]

Bug introduced in 10.0 and fixed in 10.0.2 Mathematica 10 fails to solve the following integral, saying that it does not converge. ...
7
votes
1answer
83 views

Inconsistent results for Integrate depending on irrelevant assumptions

I stumbled across this amusing issue today when doing some definite integrals: ...
7
votes
1answer
189 views

Differentiate the product of some terms

How can I compute the following derivative, $$ \frac{\partial}{\partial \lambda_j} \prod_{i=1}^k (1+\lambda_i)e^{\lambda_i} \quad \text{for }\; 1\le j \le k $$ for some positive integer $k$ which is ...
7
votes
2answers
263 views

Fundamental Theorem of Calculus for definite integrals… assume continuity?

So here's the problem: I can evaluate the indefinite integral: Integrate[D[u[x], x], x] u[x] However, I'd like to ...
7
votes
1answer
212 views

Suspected bug in Integrate

Bug introduced in 9.0.1 or earlier and fixed in 10.1.0 In version 10.0: ...
7
votes
0answers
173 views

Incorrect evaluation for Thue-Morse signed harmonic series

I would like to evaluate $$s = 1 - \frac{1}{2} - \frac{1}{3} + \frac{1}{4} - \frac{1}{5} + \frac{1}{6}+\frac{1}{7}-\frac{1}{8} - ... + \frac{(-1)^{\textrm{binary digit sum}(n-1)}}{n} + ... $$ where ...
6
votes
5answers
979 views

Find function inverse

I'm trying to find the inverse of a function: (30*x^2 (1 - x)^2) (* where 0<x<1 *) I tried all the following options: 1. ...
6
votes
2answers
2k views

Paths integrals in the complex plane

I can't find how to calculate path integrals of complex functions in the complex plane. For example: $$\oint_{\mid z \mid =2}\frac{1-e^z+z}{z^3 (z-1)^2}dz$$
6
votes
3answers
257 views

How to calculate this sum?

I want to find the sum $$S=f\left(\dfrac{1}{2012} \right) +f\left(\dfrac{2}{2012} \right) +\cdots + f\left(\dfrac{2011}{2012} \right), $$ where $$f(x) = \dfrac{4^x}{4^x + 2}.$$ I tried ...
6
votes
2answers
374 views

Convolve does not get the correct answer

Convolve[Sinc[x], Exp[-x^2], x, X] (* E^-X^2 π *) is obviously false, but why? Any suggestions ?
6
votes
3answers
305 views

Where did I go wrong with my implementation of the trapezoidal rule?

One method for doing quadrature, called the trapezoidal rule, improves accuracy by connecting the points on the curve corresponding to the points of subdivision with line segments, forming trapezoidal ...
6
votes
1answer
1k views

How to expand a function into a power series with negative powers?

Is there any way to expand this expression a+b(1-Exp[-T/(b c)]/(z-Exp[-T/ (b c)]) (where a, ...
6
votes
2answers
1k views

How to find the non-differentiable point(s) of a given continuous function?

For example, the non-differentiable point of the function $f(x)=|x|$ is at $x=0$. How to find the non-differentiable points of a continuous function that is defined numerically?
6
votes
3answers
195 views

Limit of Nested Radical $\sqrt{1+\sqrt[2!]{2^2+\sqrt[3!]{3^3+…}}}$

I want to find limit of infinite nested radical $\quad \quad \sqrt{1+\sqrt[2!]{2^2+\sqrt[3!]{3^3+...}}}$ but I don't know how to define this expression in Mathematica. How can I define it and ...
6
votes
2answers
320 views

Force integration to be linear over sum?

Is there an obvious way to force Mathematica to separately integrate the terms in a sum? According to the docs, When part of a sum cannot be integrated explicitly, the whole sum will stay ...
6
votes
1answer
386 views

What is the mathematical meaning behind D[f]?

y = a + b x; I can understand this output of the ordinary differentiation of y w.r.t. x ...
6
votes
1answer
342 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 ...
6
votes
2answers
212 views

Curvature Application

I found these amazing animations at https://courses.engr.illinois.edu/tam212/avt.xhtml#avt which I think will be wonderfully useful when teaching multivariable calculus next semester. I've started to ...
6
votes
1answer
336 views

How to solve this probability symbolically or numerically?

I am trying to calculate the following probability $$\mathbb{P} \big(\sum_{i=1}^{m} (A_i + S_i) \le L < \sum_{i=1}^{m+1} (A_i + S_i) \big)$$ where, $$A_i \sim \exp(\lambda), \quad S_i \sim ...
6
votes
3answers
182 views

Speed up derivative evaluation

I'm trying to calculate the normal vectors and the tangent vectors at the discrete points of a suface. For example: ...
6
votes
4answers
970 views

Can't compute definite integral

Consider a scalar field (in polar co-ordinates), $f(r) = l-r$. Now I want to evaluate the field integral over a circular region of radius $b$, centered at a distance of $x$ from the origin. According ...
6
votes
1answer
159 views

Defining a function implicitly and calculating its derivatives

I'm interested in using Mathematica's symbolic manipulation to obtain, for a particular function $f$, derivatives of arbitrary order evaluated at zero. Normally I'd use the ...
6
votes
1answer
230 views

Find exponential generating function from the first few terms

The function FindGeneratingFunction computes the ordinary generating function of a sequence, given a sufficient number of initial terms. I have two questions in ...
6
votes
1answer
212 views

How to compute the residue of $e^{z-\frac{1}{z}}$ at z=0?

I've tried the following but it didn't work: Residue[Exp[z - 1/z], {z, 0}] not even this: Residue[Exp[1/z], {z, 0}] ...
6
votes
2answers
185 views

Series resulting in “No more memory available.”

Is there any way to force Mathematica to come up with the closed form to Sum[1/(i^18 (i^2 + j^2)), {i, 1, Infinity}, {j, 1, Infinity}] or ...
6
votes
1answer
109 views

Why is RegionMeasure so slow when calculating intersection area of a 2D and a 3D object?

If you think this description is too long, you can read the problem directly I know normally when one wants to calculate an region, this guide is useful. However, when it comes to calculating an ...
6
votes
2answers
243 views
6
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2answers
240 views

Is there a built-in function which detects singularities in a function?

Given a function f[x] and a region M in the complex x-plane, how can I find singularities of f in this region, i.e., issue a ...
6
votes
3answers
315 views
6
votes
1answer
119 views

Traces of the level surface $z=4x^2+y^2$

I came up with this method to plot the traces of the surface $z=4x^2+y^2$, in this case for $z=1$, 2, 3, and 4. ...
6
votes
1answer
183 views

Working with a system of differential equations that cannot be solved explicitly

I have to work a lot with three functions $\;o_1(t), o_2(t), o_3(t)\;$ that are solutions to the certain system of differential equations: ...
6
votes
1answer
91 views

Is it possible to find a function from first few terms in the expansion

Is it possible to find a function if first few terms of the expansion is known. For example if I have this series $f(x)=\frac{k^3 x^2}{6}-\frac{k^5 x^4}{120}+\frac{k^7 x^6}{5040}-\frac{k^9 ...
6
votes
1answer
963 views

Convolving/integrating problems

Let's say I want to convolve two functions (f and g), a gaussian with a breit-wigner: ...
6
votes
1answer
214 views

Solving $\frac{dx}{dt} = A \frac{ (1-x)}{(t-t^2)} - \frac{(B*x -C*x^2)}{(t-t^2 )*(t-x)}$

I would like to solve the following equation: $\frac{dx}{dt} = A \frac{ (1-x)}{(t-t^2)} - \frac{(B*x -C*x^2)}{(t-t^2 )*(t-x)}$ I have already posted it here, but still it doesn't work for me (I ...
6
votes
1answer
225 views

Incorrect evaluation of integral involving a DiracDelta, whose argument has infinitely many zeros

Let's say $X>0$ is a random variable with probability density $p_X(x)={\rm e}^{-x}$. Define the random variable $Y=\sin(X)$. From the transformation theorem for probabilities we know that its ...
6
votes
1answer
123 views

Is it possible to circumvent a bug inside SeriesCoefficient?

As far as I can tell, there seems to be a bug in SeriesCoefficient: ...
6
votes
1answer
82 views

Examining the function $f(x,y)=xy(x^2-y^2)/(x^2+y^2)$

Consider the function $$f(x,y)=\begin{cases} xy\dfrac{x^2-y^2}{x^2+y^2},&(x,y)\ne(0,0)\\ 0,&(x,y)=(0,0) \end{cases}$$ I can show that $f_x(0,0)=f_y(0,0)=0$. ...
6
votes
1answer
251 views

Integral of x^p

Can anyone explain why Mathematica does not return a conditional expression that handles the case of p=-1 for Integrate[x^p,x]? ...
6
votes
4answers
338 views

Conditional Expectation — How can Mathematica find a more general closed form?

Mathematica can't find a solution to this Expectation. ...