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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7
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2answers
1k 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
1answer
527 views

Integral of the Sinc product

Let as consider the following integral $$ B_n = \int_0^\infty \prod_{k=1,3,5,\dots}^n\frac{\sin (x/k)}{x/k}dx $$ By definition, Sinc[x]=Sin[x]/x therefore ...
7
votes
2answers
280 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
859 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
464 views

Double integral over a parallelogram

Problem: Evaluate the following double integral $$\int \!\!\! \int_D (14 x^2+61 xy+42 y^2)^{3} \, dxdy$$ where $D$ is a parallelogram between these four lines: $2x+7y=6$, $2x+7 y=-6$, $7x+6 y=6$ and ...
7
votes
1answer
651 views

Hankel Transform integrals won't work in Mathematica

I'm trying to do this integral, which is shown on the Wikipedia page on the Hankel transformation: $$\int_0^{2\pi}\mathrm d\varphi\;e^{\mathrm im\varphi}e^{\mathrm ikr\cos(\varphi)}$$ The answer is ...
7
votes
2answers
253 views

Definite integral incorrectly giving a nonreal value

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.) Why? This arose when I ...
7
votes
2answers
301 views

Find asymptotics of Sum[2^i*Binomial[n-i-1,2*n/3-1],{i,0,n/3}]

I have an expression 2^n / Sum[ 2^i Binomial[ n - i - 1, 2n/3 - 1], { i, 0, n/3}] ...
7
votes
1answer
656 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
422 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
253 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
685 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
436 views

Limit not evaluating

Why is the input Limit[IntegerPart[Sin[x]/x], x -> 0] not being evaluated?
7
votes
1answer
183 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
1answer
431 views

Numerical contour integrations in the complex plane - contour deformation gives different answer for analytic kernel

I am trying to do a contour integration in Mathematica numerically. In particular, I'm checking the identity: $$ H_m^{(1)}(z) =\frac{i^{-m}}{\pi}\int_{-\pi/2 + i \infty}^{\pi/2 - i \infty} \exp[i m ...
6
votes
5answers
508 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
3answers
254 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
3answers
271 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
937 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
271 views

About an infinite product

I have a curiosity as regards the infinite product below. I wonder why Mathematica v.8.0. says the limit is $1$. This is not true. ...
6
votes
1answer
6k views

Quick Hessian matrix and gradient calculation?

I am absolutely new to Mathematica and I actually want to try implementing a little optimization method . Long story short assuming I have a predefined two-variable function f(x,y) I want to ...
6
votes
2answers
268 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
2answers
5k views

Simple ways to evaluate a derivative at a point?

The contrast in behavior between, say, f[x_] = Sin[x^2]; f'[2] vs. u[x_, y_] = Cos[x + y^2]; has always bothered my ...
6
votes
1answer
376 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
316 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
1answer
316 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. ...
6
votes
1answer
247 views

solve ODE with divergencies

The solutions of the second order differential equation $$\frac{1}{\eta}\frac{d}{d\eta}\left(\eta \frac{df}{d\eta}\right)+\left(1-\frac{s^2}{\eta^2}\right)f-f^3=0$$ is shown in Fig. 5.2 below, for ...
6
votes
4answers
865 views

How do I obtain the correct double limit?

The command Limit[(Sin[x^2] + Sin[y^2])/ (x - y) /. x -> 0, y -> 0] (* 0 *) I think that Mathematica finds the iterated limit instead of the double ...
6
votes
4answers
900 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
2answers
175 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
144 views

What exactly does GenerateConditions do?

Consider for example this strange behavior: Integrate[1/x, {x, 0, Infinity}, GenerateConditions -> False] (*0*) I'd also like to know the difference between ...
6
votes
2answers
198 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
1answer
79 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
181 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
116 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
225 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
273 views

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

Mathematica can't find a solution to this Expectation. ...
6
votes
1answer
62 views

Inconsistent results for Integrate depending on irrelevant assumptions

I stumbled across this amusing issue today when doing some definite integrals: ...
6
votes
2answers
209 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 ...
6
votes
1answer
153 views

Problem overloading D[] using UpValues

First I define an entry of UpValues overloading D[] for expressions with head ftest: ...
6
votes
0answers
147 views

Suspected bug in Integrate

Bug introduced in 9.0.1 or earlier and persisting through 10.0.2 or later In version 10.0: ...
6
votes
0answers
159 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 ...
5
votes
4answers
1k views

How to find the sum all even numbers of this sequence?

I have a sequence $(u_{n})$ $$u_1= 1, \quad u_2 = 2, \quad u_3 = 3, \quad u_{n}= -u_{n-3} + 3u _{n-2} +2 u_{n-1}, \quad \forall n \geqslant 4.$$ I want to list the first $20$ terms of this sequence ...
5
votes
3answers
292 views

How to force Dt[ ] to recognize dependencies

I want to differentiate a function f[] for which I don't have a specific expression. f[] depends on ...
5
votes
3answers
782 views

recursive integration

I am trying to do multiple integrations recursively. For instance, I would like to do the following equation for arbitrary integer $n$: $\displaystyle R_n(t) = \int_0^t \mathrm dt' R_0(t-t') ...
5
votes
2answers
344 views

How can I calculate the limit without using the L'Hopital's rule

I need to prove this limit without using the L'Hopital's rule: $$\lim_{x\to 0} \frac{(1+a\,x)^{1/4} - (1+b\,x)^{1/4}}{x} = \frac{a-b}{4}$$ How can I do it in Mathematica?
5
votes
3answers
351 views

Bug in complicated Limit

The code Limit[Log[2 - Sin[x]*Cos[x]], x -> Infinity] outputs Interval[{0, Log[3]}] in Mathematica 10.0.2.0 . It should ...
5
votes
2answers
941 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?
5
votes
2answers
565 views

How to receive the result of the integral $ \int\frac{1}{x}dx$ is $\ln|x|$?

When I input \[Integral]1/x \[DifferentialD]x in Mathematica, I get Log[x]. How can I make the result of the integral ...
5
votes
2answers
179 views

Unexpected Integration Constant [closed]

Why does this integration Integrate[15-30x+6x^2-1/(x+5),x] return 700 as integration constant? If we make a slight modification ...