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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How can you compute Itō Integrals with Mathematica?

How can you compute Itō Integrals with Mathematica? I tried searching through the documentations but I didn't find anything. P.S. I was not at all sure how to tag this question. I had to put in at ...
12
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2answers
7k views

How to specify assumptions before evaluation?

If I request mathematica evaluate an integral for me, I'll often get a more general ConditionalExpression than I want. Example : ...
12
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2answers
1k views

Suppressing negative roots in Mathematica

Problem Using Mathematica's Solve operator can sometimes lead to an output involving a positive and negative root (say when solving for a variable such as ...
12
votes
3answers
376 views

How can I obtain an asymptotic integral expansion at infinity?

I want to find the asymptotic expansion at $x \to \infty$ of the following function: $$ I(x) \equiv \int_0^{\pi/2} e^{-xt^3 \cos(t)} dt.$$ To do this I defined ...
12
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1answer
330 views

Why is this infinite series wrongly computed by Mathematica?

Could you let me know if Mathematica (newer versions) is able to correctly compute this one? Sum[(-1)^(n + 1) Cos[3^n x]^3/3^n, {n, 1, Infinity}]
11
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4answers
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Finding critical points of a function

Let's say we'd like to find the critical points of the function $f(x)=\sqrt{x-x^2}$. Finding out where the derivative is 0 is straightforward with Reduce: ...
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3answers
2k views

how to simplify symbolic integration

The following is the code: ...
11
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3answers
710 views

Determine frequency of oscillations

I am wondering how I could determine the frequency of oscillations of a differential model equation? How could I find the frequency from this example given in Mathematica Documentation: ...
11
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3answers
254 views

How to efficiently find moments of a multinormal distribution?

I have a multinormal distribution with covariance matrix $\sigma$ and zero mean. I want to find moment $$ E[x_1^{r_1}x_2^{r_2}\cdots x_n^{r_n}] =\,? $$ Of course, there is a build-in function ...
11
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2answers
4k views

Equation of a line that is tangent to a curve at point

A common problem in the derivative section of calculus texts is "find the equation of the line that is tangent to the curve $y = \ldots$ at the point $P$." To find the line that is tangent to $y = 2 ...
11
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6answers
5k views

Finding the centroid of the area between two curves

When I have an area bounded by curves, is there a built-in way to find the center of the area? Or do I have to plot it first and then use ComponentMeasurements on ...
11
votes
2answers
688 views

Series expansion of an inverse

I have to find the series expansion of the inverse function of : $\arctan\left(\frac{\ln(1+x)}{1+x}\right)$ How do I find out the series expansion of any inverse ? Note: The inverse of a function ...
11
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3answers
372 views

Find asymptotics of $\sum\limits_{i=0}^{n/3} 2^i \binom{n-i-1}{\frac{2n}{3}-1}$

I have an expression 2^n / Sum[ 2^i Binomial[ n - i - 1, 2n/3 - 1], { i, 0, n/3}] ...
11
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2answers
1k views

Maximizing a function with assumptions

Using f[s_] := Log[(s/r)^α ((α - 2) n0 r^α + 2 π Pmax ρ r^2) /((α - 2) n0 s^α + 2 π Pmax ρ s^2)]/s When I run the following line: ...
11
votes
1answer
183 views

Keeping Integrate from making unnecessary assumptions

I would like to evaluate the integral $\int_{-\infty}^\infty \mathrm{d}x \, \exp\left(- a x^2 - x^4\right)$ for any real value of $a$. Mathematica 8.0.4 gives the following result: ...
10
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8answers
1k views

Finding maximum or minimum of implicit functions

is there any built in function that can be used to find maximum or minimum of implicit functions? For example, if we have the equation $$x^2 + y^2 = (2 x^2 + 2 y^2 - x)^2,$$ then we can visualize the ...
10
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3answers
788 views

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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4answers
4k views

How can I differentiate Numerically?

Mathematica has two ways to integrate: Integrate and NIntegrate. But what about D? ...
10
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6answers
707 views

Integrate gives wrong results

Integrate[a/(Sin[t]^2 + a^2), {t, 0, 2 Pi}] $$\int_0^{2 \pi } \frac{a}{a^2+\sin ^2(t)} \, dt$$ gives $0$ This cannot be true. What is going on? If I insert a ...
10
votes
2answers
716 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 ...
10
votes
4answers
365 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 ...
10
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2answers
663 views

Why does Integrate declare a convergent integral divergent?

When I try this command Integrate[1/Sqrt[(s^2 - u)^2 - 1], {s, m, Infinity}, Assumptions -> u > 2 && m > 10] Mathematica declares that the ...
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2answers
136 views
10
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1answer
480 views

Why does Mathematica return a Fourier transform for a function for which it is not defined?

The following function $$g(x) = (1 + x^{1/a} )^a $$ should NOT have a Fourier transform, as far as I am aware, for any real values of $a$ since $g(x)$ is not nice in the sense of decays quickly ...
10
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1answer
446 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
390 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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2answers
238 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 ...
10
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2answers
381 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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2answers
238 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 ...
10
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3answers
372 views

Typeset unbalanced brackets

In LaTeX, you can typeset expressions containing unbalanced brackets, by balancing them with an invisible delimiter \left. or ...
9
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4answers
1k views

Plot a 2D vector path onto a surface

In my calculus 3 course, we're studying gradients and have a project that takes a combination of 3D Gaussian radial surfaces and a basic parametric path $r(t) = \{x(t),y(t)\}$ to see how the gradient ...
9
votes
4answers
2k views

How to plot and find the volume of a solid?

How to plot and find the volume of the solid enclosed between the paraboloid z=5(x^2+y^2) and z=6-7x^2-y^2 And the answer of ...
9
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3answers
381 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 ...
9
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2answers
208 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 ...
9
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2answers
741 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 ...
9
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3answers
759 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
654 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) + ...
9
votes
2answers
3k 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 ...
9
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2answers
635 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 ...
9
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2answers
365 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: $$V_0(r)=2r$$ ...
9
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4answers
342 views

Mathematica complaints that convergent integral diverges

Bug introduced in 10.0.0. Fixed in 10.0.2 Trying to do the integral ...
9
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1answer
230 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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2answers
354 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 ...
9
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1answer
302 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 ...
9
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1answer
286 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
180 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 ...
9
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1answer
333 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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0answers
226 views

Is there a way to teach integrate new solutions?

I have an integral which I can solve, but integrate cannot: ...
8
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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
712 views