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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14
votes
1answer
213 views

Is it a bug in Limit?

Bug introduced in 9.0 and fixed in 10.0.0 Mathematica 9 on Windows and WolframAlpha says Limit[Sin[Sqrt[x+1]]-Sin[Sqrt[x]],x->Infinity] is ...
14
votes
1answer
269 views

Why do big-O terms disappear in definite integrals since Mathematica 9?

In Mathematica 8, when I computed the following input: Integrate[Series[Cos[x], {x, 0, 2}], {x, 0, a}] Mathematica returned an expression that had a O[a^4] in ...
14
votes
0answers
279 views

Why does Mathematica choose branches as it does in this situation?

Consider these integrals: ...
13
votes
9answers
2k 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 ...
13
votes
5answers
807 views

Does $x>0$ imply that $x\in\mathbb{R}$?

Let’s assume I input Assuming[x > 0, expression] Is it assumed by Mathematica that $x$ is a real number? Or that the real part of $x$ is positive? Something ...
13
votes
2answers
448 views

What kind of hypergeometric function is it?

I found a formula for an integral of a product of three Bessel functions at The Wolfram Functions Site: I cannot understand what kind of hypergeometric function it is. The Mathematica code given ...
13
votes
1answer
427 views

What strategies can I use to evaluate a limit when Limit[] returns unevaluated

I'm trying to find the following limit using Mathematica: $$\lim_{N\to\infty}\sum_{k=1}^N\left(\frac{k-1}{N}\right)^N$$ The problem is taken from here and is known to converge to ...
13
votes
1answer
1k views

Plot showing discontinuity where it shouldn't

I was trying to integrate a continuous function with a kink and I did it two ways and both ways the plot of the result shows a discontinuity. I also later want to differentiate the Integrated ...
13
votes
1answer
378 views

How do I get a general form for n-th derivative of a function? [duplicate]

I tried the following but it basically returns unevaluated: ...
13
votes
1answer
468 views

Why is this infinite series wrongly computed by Mathematica?

Bug introduced in 7.0 and fixed in 10.0 Could you let me know if Mathematica (newer versions) is able to correctly compute this one? ...
13
votes
1answer
320 views

Extra factors appear when evaluating Euler integrals

Note: this is fixed in version 9. When I perform the double integral in Mathematica, Integrate[(x (1 - x))^z (y (1 - y))^z, {x, 0, 1}, {y, 0, 1}] which ...
13
votes
1answer
443 views

Why does Mathematica give an incorrect answer to this multiple integral?

This is not a new problem but I would like to understand why Mathematica gives the result that it does. (Volume of a hypersphere) ...
12
votes
5answers
769 views

Double series over primes

I'm very curious if the following double series over primes has a closed form: $$\sum_{k \in \mathcal{P}}\sum_{n \in \mathcal{P}}\frac{1}{k\;n(k+n)^2}$$ where $\mathcal{P}$ denotes the set of all ...
12
votes
6answers
6k 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 ...
12
votes
6answers
480 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 a ...
12
votes
5answers
2k views

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
votes
3answers
514 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
votes
2answers
9k 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
votes
2answers
355 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 ...
12
votes
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
1answer
882 views

1/x integration giving 0 instead of infinity [duplicate]

I stumbled across the fact this integral: Integrate[1/y, {y, 0, 1}, GenerateConditions -> False] returns 0 Anybody ...
12
votes
1answer
362 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 ...
12
votes
2answers
399 views

Why Mathematica cannot apply The Fundamental Theorem of Calculus automatically?

For some reason Mathematica cannot evaluate this definite integral: ...
12
votes
1answer
163 views

Why do singularities disappear when performing convolutions with the Dirac Delta function?

I am attempting to perform a convolution involving the Dirac Delta function: $\int_{-\infty}^{\infty} \frac{1}{t+1} \cdot \delta(t+1)\ dt$ I would expect that the result of this integral is ...
11
votes
4answers
2k 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 ...
11
votes
4answers
8k views

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: ...
11
votes
3answers
951 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
votes
3answers
2k views

how to simplify symbolic integration

The following is the code: ...
11
votes
2answers
5k 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
votes
2answers
783 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
votes
2answers
787 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 ...
11
votes
3answers
389 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
votes
2answers
965 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 ...
11
votes
1answer
228 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 ...
11
votes
2answers
2k 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
191 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: ...
11
votes
3answers
419 views

Typeset unbalanced brackets

In LaTeX, you can typeset expressions containing unbalanced brackets, by balancing them with an invisible delimiter \left. or ...
10
votes
3answers
933 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 ...
10
votes
6answers
789 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
4answers
5k views

How can I differentiate Numerically?

Mathematica has two ways to integrate: Integrate and NIntegrate. But what about D? ...
10
votes
3answers
433 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 ...
10
votes
2answers
409 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
votes
2answers
817 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
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 ...
10
votes
2answers
215 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. ...
10
votes
4answers
421 views

Mathematica complaints that convergent integral diverges

Bug introduced in 10.0 and fixed in 10.0.2 Trying to do the integral ...
10
votes
2answers
866 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} ...
10
votes
2answers
408 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: $$V_0(r)=2r$$ ...
10
votes
1answer
370 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. ...
10
votes
2answers
465 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 ...