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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3answers
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Different results for integration using Mathematica and MATLAB

I have the following integration: $$\text{y}=2 \sqrt{\frac{1}{\pi }} \int_0^{\infty } \frac{e^{-z} \left(1-e^{-\frac{z}{b}} \left(\frac{a}{a+c z}\right)^L\right)}{\sqrt{z}} \, dz$$ I get different ...
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3answers
7k 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 $n = 3$ ...
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3answers
473 views

Maximize violating constraints

I have Maximize[{(h*10)/(300*(100 - (l^.5 + d^.4 + H^.6))), (l + d + H + h) == 669, l > 0, d > 0, H > 0, h > 0}, {h, l, d, H}] I believe ...
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2answers
423 views

Symbolic integration fails while numerical integration succeeds

I am hoping to evaluate the following integral Integrate[((r^3 - 7)^(2/3)*(1 - (r^3 - 7)^(2/3)/r^2))/r^3, {r, 2, Infinity}] but Mathematica informs me that this ...
10
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2answers
251 views

Volumes of Revolution with detailed diagram

All, In calculus I have to do images such as the following in helping explain technique to students. This one is by rotating $y=\sqrt x$ about the x-axis, an image copied from Stewart's Calculus ...
10
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2answers
921 views

Find closed form expression for series expansion coefficients [duplicate]

Is there a built-in function that will find a general expression for the coefficient of the series expansion of a function? Series will only give the explicit ...
10
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2answers
866 views

Derivative from the left and right

I am currently trying to write a script that takes a curve $C$, a starting point $p$ and a direction vector $\vec{v}$ and calculates the path of a beam starting at $p$ in direction $\vec{v}$ when it ...
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2answers
4k views

How do I evaluate a double or triple integral over a region?

Say I need to evaluate the integral $\iiint_W f(x,y,z) dx dy dz$ and $W$ is a region given to me like $W = \{ (x,y,z) : 1 \leq x^2 + y^2 \leq 4, 1 \leq z \leq 5\}$. I don't how to do this with a ...
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2answers
225 views

Demonstrating the behavior of a function as its independent variable approaches zero

I have several questions regarding the function $$f(x)=\frac{\sqrt{x^2+9}-3}{x^2}$$ that I would like to help my students with in the upcoming semester. Now, the limit as $x\to 0$ is 1/6. ...
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4answers
452 views

Mathematica complaints that convergent integral diverges

Bug introduced in 10.0 and fixed in 10.0.2 Trying to do the integral ...
10
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2answers
449 views

Recursive Integral for Volume of $n$-Ball

The volume of an $n$-ball (the $(n+1)$-dimensional analogue of a disk) of radius $r$ can be found by the following integral recurrence: $$V_0(r)=2r$$ $$V_n(r)=\int_{-r}^rV_{n-1}\left(\sqrt{r^2-x^2}\...
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2answers
887 views

Is Mathematica really getting this limit wrong?

Bug introduced in 5.1 and fixed in 10.0.2 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} ...
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1answer
382 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. ...
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2answers
535 views

Interpolating an Antiderivative

I'd like to be able to make InterpolatingFunctions for antiderivatives of functions that can't be integrated symbolically. However, the following code returns ...
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2answers
155 views
10
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2answers
333 views

Asymptotics of $\frac{\sum _{i=0}^{\lfloor n/2 \rfloor} {2(n-2i) \choose n-2i} {n \choose 2i} {4i \choose 2i}}{2^{3n - 1}}$

I am fairly sure that asymptotically $$\frac{\sum _{i=0}^{\lfloor n/2 \rfloor} {2(n-2i) \choose n-2i} {n \choose 2i} {4i \choose 2i}}{2^{3n - 1}} \sim \frac{2}{\pi n}.$$ I tried Limit[n*Sum[...
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1answer
251 views

Vastly incorrect answers obtained by increasing WorkingPrecision with modified Bessel functions

Bug introduced in 7.0 or earlier and persisting through 10.4.1 This is a follow-up to this question regarding numerical instabilities occurring with modified Bessel functions. In trying to explore J....
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1answer
1k views

How does a Pringle lose its curvature?

Nom! As part of a bigger project, I've was writing some code to calculate the scalar curvature of surfaces of the form $z = f(x,y)$. This uses a general calculation of the scalar curvature to produce ...
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1answer
561 views

Is it possible to find a limit of a sequence given by its recurrence relation?

I need to calculate a limit of a sequence given by its recurrence relation. I tried the following: ...
10
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1answer
484 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 ...
10
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1answer
358 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 ...
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1answer
479 views

Compute inverse Laplace transform with Integrate

Since inverse Laplace transform is just an integral, while we already have InverseLaplaceTransform built in, can we compute it with ...
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1answer
297 views

Contour Integration along a contour containing two branch points

I need to compute following contour integrations: $$f(u)=\oint_\alpha dz \sqrt{z^3+z+u} \qquad ; \qquad g(u)=\oint_\beta dz \sqrt{z^3+z+u}$$ In which $\alpha$ and $\beta$ are two contours in ...
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2answers
268 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 ...
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2answers
565 views

How to take derivative of parameterized coordinate?

Suppose I have a vector in $\mathbb{R}^n$ but $n$ is not known in advance. I want to be able to write functions which operate on the components of that vector, and then I'd like to be able to take ...
10
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1answer
108 views

Slot number cannot be filled

I got a strange message with the derivative of a two-argument function defined by a definite integral. For example if "f" is defined as: ...
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1answer
303 views

Dirichlet coefficients as limits: wrong

Perhaps I should have included the word "bug" in my question. Here we go There is a bug in this Limit Limit[3^s (-1 - 2^-s + Zeta[s]), s -> ∞] (* 0 *) which ...
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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 <...
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0answers
161 views

Strange behaviour of integrals with Cos, Sin, and Exp

During the study of the problem How to solve this integration? I have discovered a strange behaviour of some integrals. I would consider it a bug. ...
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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 ...
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5answers
6k views

Notation of partial derivative

I want to write partial derivatives of functions with many arguments. Why is it that when I type f[x,y] ctrl+6 (0,1) it turns out to be bad syntax? The output of ...
9
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2answers
3k 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: $$\oint_{\mid z \mid =2}\frac{1-e^z+z}{z^3 (z-1)^2}dz$$
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6answers
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4answers
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Is it possible to “pretty print” my input integral in Mathematica like Wolfram|Alpha does?

With Wolfram|Alpha, if I enter: Integrate[r^3, {t, 0, 2Pi}, {r, 0, 2}, {z, r^2, 4}] Wolfram|Alpha will pretty print my input like so: Which is very readable, ...
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3answers
915 views

Inverting a function in a certain region

InverseFunction works well for globally invertible functions, like f = 2*# + 2 &; InverseFunction[f] ...
9
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2answers
779 views

Why aren't these additions of integrals and summations equal?

I have the following code: Simplify[Integrate[f[x] + g[x], x] == Integrate[f[x], x] + Integrate[g[x], x]] To test: $$\int{\left(f(x) + g(x)\right)dx}=\int{f(x)...
9
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2answers
992 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 ...
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2answers
750 views

Problem with numerical evaluation of analytically solved integral, solution way off

The following command in Version 9.0.1: N[Integrate[x^50*Sin[x], {x, 0, 1}]] gives $1.4615\times 10^{48}$ which is way off from the correct solution which is ...
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2answers
284 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 $\int_0^1\int_0^1\int_0^1\frac{1}{xyz}\,dz\,dy\,dx=0$. ...
9
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1answer
440 views

Laplacian and DiracDelta

It is known that: $\nabla^2 \dfrac{1}{|r-r'|} = - 4 \pi \delta^3(\vec{r}-\vec{r}')$. If I do that with Mathematica, I find: ...
9
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1answer
919 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 ...
9
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1answer
167 views

When to use GenerateConditions -> True

Many functions, usuallly those involving integration, take a GenerateConditions option which often defaults to False, or at ...
9
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1answer
305 views

Mathematica: computing a difficult integral

I am trying to compute the following integral: Integrate[Exp[Sum[-((cw λ - b[i])^2/(2 σ^2)), {i, 1, n}]], {cw, 0, 1}] And currently Mathematica outputs ...
9
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1answer
411 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
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3answers
453 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?
9
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1answer
263 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 ...
9
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1answer
750 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 $n \times 1$ ...
9
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1answer
140 views

Non-rectilinear integration of InterpolatingFunction

Bug introduced in 7 or earlier and persisting through 10.3.1 [CASE:3487737] I have a 2-d array that I would like to resample into a different coordinate system and integrate along one of the new ...
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1answer
165 views

Wrong limit: Limit[(1 + (-1)^n/n)^n, n -> Infinity]=1 (Mathematica 10.4 and W-Alpha)

Since $(1-\frac{1}{n})^n\to 1/e$ and $(1+1/n)^n\to e$, the sequence $(1+\frac{(-1)^n}{n})^n$ has no limit as $n\to\infty$, but has limits for odds and even numbers. If $n$ were taken to be real, there ...