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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9
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1answer
339 views

How to calculate this integral? Integrate[BesselJ[0, x - BesselJZero[0, 1]]/x, {x, -Infinity, Infinity}]

I tried to calculate the following integral, but it returned unevaluated. ...
9
votes
1answer
193 views

Why doesn't Mathematica evaluate this simple limit?

I want to evaluate $$\displaystyle\lim_{n\to\infty}\left(n-\sqrt{\sin(n)+10n+n^2}\right)^2$$ I used this code ...
9
votes
0answers
228 views

Is there a way to teach integrate new solutions?

I have an integral which I can solve, but integrate cannot: ...
8
votes
3answers
2k views

Using D to find a symbolic derivative

I need to do the following: Define a function Take the derivative of this function and have a look at the symoblic representation Substitute in some values With the bonus that I want to use the ...
8
votes
5answers
719 views
8
votes
5answers
551 views

simplifying $\frac{\log x^a}{a} = \log x$

If one makes the assumptions $x>0,a>0$, then $\frac{1}{a}\log x^a = \log x$. Thus, in Mathematica, Simplify[1/a*Log[x^a], {a > 0, x > 0}] returns ...
8
votes
3answers
5k views

How to find the nth derivative?

This question is not the same as my last one. How do you find the $n$-th derivative where $n$ is a variable? For example, you can find the nth derivative for a specific $n = 3$ ...
8
votes
2answers
2k views

How do I find line integrals?

For example, how can I calculate $$\int_{\left | z \right |=1}\frac{dz}{z}$$ I know that the answer is $2\pi i$ but how do I do it using Mathematica?
8
votes
3answers
592 views

How can I calculate the perimeter of an equation-defined curve?

This is how the curve looks like: ...
8
votes
2answers
526 views

Weird plot with SphericalPlot3D

Taking the equation $x^2-y^2-z^2=1$ and using ContourPlot3D: ContourPlot3D[ x^2 - y^2 - z^2 == 1, {x, -3, 3}, {y, -3, 3}, {z, -3, 3}] Yields the proper image. ...
8
votes
2answers
316 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. ...
8
votes
3answers
437 views

Biot-Savart Law/Magnetostatics Solution

I am working on solving the Biot-Savart Law equation for the magnetic field around a charged ring of uniform current density. The expression that Mathematica gives is rather nasty, as expected. Notice ...
8
votes
4answers
368 views

Definite integral incorrectly giving a nonreal value

fixed in 10.1 (windows and Mac OS X) 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 ...
8
votes
2answers
763 views

Is Mathematica really getting this limit wrong?

I'm trying to calculate: $$ \lim_{\beta \to \infty} \tanh \left( \beta A \right) = \mathrm{sgn} \left( A \right) $$ for $A \in \mathbb{R}$, $\mathrm{sgn} \left( \# \right)$ being the sign function. ...
8
votes
4answers
1k 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 ...
8
votes
1answer
758 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 ...
8
votes
2answers
597 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 ...
8
votes
1answer
651 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 ...
8
votes
1answer
353 views

ReplaceAll[] and Limit[] don't give correct results for this expression under extreme variables [duplicate]

Possible Duplicate: Funny behaviour when plotting a polynomial of high degree and large coefficients 1/x^2 + (3 + x)/(6 (1 - Exp[x] + x)) ——This is a ...
8
votes
2answers
238 views

Make Sinc'[0] return 0 instead of Indeterminate

I want Sinc'[0] to return 0, but instead it returns Indeterminate. I've tried Unprotect[Sinc] Unprotect[Derivative] Derivative[1][Sinc][0] ^= 0 But it doesn't ...
8
votes
1answer
155 views

Curious solutions of x' = Sqrt(x), x(0)=4

Consider the solution of $x'=\sqrt{x}$, $x(0)=4$ using DSolve. ...
8
votes
2answers
298 views

Why Mathematica cannot apply The Fundamental Theorem of Calculus automatically?

For some reason Mathematica cannot evaluate this definite integral: ...
8
votes
1answer
228 views

Expanding derivatives of hypergeometric functions

Sometimes Mathematica expresses results of integration or summation in terms of symbolic derivatives of Hypergeometric2F1 function, and cannot further simplify ...
8
votes
2answers
741 views

Symbolic integration in real domain only ( assumptions and ComplexExpand don't work)

Integrate[m^2/((x - m^2)^2 + y^2), m] mathematica gives me a complex-valued reuslt, but maple 17 gives me what I want. I tried using assumptions, but it doesn't ...
8
votes
1answer
529 views

Limit not evaluating

Why is the input Limit[IntegerPart[Sin[x]/x], x -> 0] not being evaluated?
8
votes
1answer
232 views

Why don't products of Dirac deltas integrate correctly?

Bug introduced in 10.0.0 and fixed in 10.0.1 The integral $\int \int \ \delta(x) \delta(y) \ dx dy=1$ evaluates to 0 in Mathematica ...
8
votes
3answers
313 views

Pulling constants out of integrations [duplicate]

The following is the code and output of a Mathematica command How do I get Mathematica to remove $g(y)$ outside the integral?
8
votes
1answer
624 views

Is there any automatic differentiation package?

I'm wondering if an automatic differentiation package exists for Mathematica. This is what I mean by automatic differentiation.
8
votes
1answer
133 views

LogisticSigmoid residue bug?

Bug introduced in 10.0 and fixed in 10.2.0 On version 10.0 for arm: Residue[LogisticSigmoid[z],{z,I Pi}] gives 0 not 1. ...
8
votes
2answers
890 views

Bug in Integrate for Mathematica

Consider the following: ...
8
votes
1answer
572 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 ...
8
votes
1answer
511 views

Integral with HeavisideTheta takes too long to evaluate

I tried to compute $$\int_{-1}^1 d x_1 \int_{-1}^1 d x_2 \int_{-1}^1 d y_1 \int_{-1}^1 d y_2 \theta(x_1 x_2 + y_1 y_2)\,$$ where $\theta$ is Heaviside's step function, by using ...
8
votes
0answers
57 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 ...
7
votes
6answers
1k 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 ...
7
votes
3answers
340 views

Generating a polynomial that's accurate to within an error of no more than 1/10^5

I'm currently stuck on a question for class that asks... "Find a polynomial p[x] that you can use to calculate 6 ArcTan[x] to ...
7
votes
2answers
522 views

Probability: Calculating a multiple integral

Find the value of $P[\Pi_{i=1}^{10}X_i > C]$ for $C=2,5$, where $X_{10\times 1}$ is a random vector with $10$ dimensional Cauchy Distribution having location parameter $\mu_{10\times 1} = ...
7
votes
3answers
330 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 ...
7
votes
2answers
661 views

Simplifying the derivative of $|x|$

Context In[855]:= D[Abs[x], x] /. x -> 1 Out[855]= Derivative[1][Abs][1] In[856]:= D[x, x] /. x -> 1 Out[856]= 1 Question Why is ...
7
votes
1answer
8k 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 ...
7
votes
4answers
1k views

Mathematica gives wrong answer for an integral

When I execute the following, Integrate[ Exp[-w^2 + I w^3], {w, -∞, ∞}] I get (2 E^(2/27) BesselK[1/3, 2/27])/(3 Sqrt[3]) ...
7
votes
2answers
6k 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 ...
7
votes
2answers
226 views

Why does Mathematica say $\int_0^1\int_0^1\int_0^1\frac{1.0}{xyz}\,dz\,dy\,dx=0$?

Mathematica 9 says that $\int_0^1\int_0^1\int_0^1\frac{1.0}{xyz}\,dz\,dy\,dx=0$ and $\int_0^1\int_0^1\int_0^1\frac{1}{xyz}\,dz\,dy\,dx=0$. ...
7
votes
1answer
329 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
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
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
795 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
470 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
286 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
780 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 ...