# Questions tagged [calculus-and-analysis]

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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### 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 ...
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### Calculating a limit with a result that is discontinuous in the parameters

The following limit is left unevaluated (Edit: added the assumption that $\epsilon$ is real thanks to the comment below): ...
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### How to deal with complicated Gaussian integrals in Mathematica?

As we know, for most Gaussian integrals, we can get the analytical result. Now I have many Gaussian integrals to treat, which have the following general form, ...
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### How to substitute numeric values in a symbolic Jacobian matrix?

I have a multi-variate function from $\mathbb{R}^n\to\mathbb{R}^n$. Choosing any desired initial vector, we can produce the corresponding function value, which is a vector as follows. The main problem ...
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### How does Mathematica integrate?

Basically, this question can be considered to be an extenstion to my other question. What I wanted to do was this integral as homework (it is indefinite BTW so no approximations using Simpson's Rule ...
947 views

### Quickly differentiate and evaluate a function of several variables

How can I differentiate a function with respect to several variables, and evaluate it at the same time? I want to specify also the variable index that I want to differentiate, and the number of times ...
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### 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 I'...
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### Integrating with multiple indicator functions

I am trying to calculate an integral involving multiple indicator functions, such as: $$h(u,v,w) = -\int_0^1 J^{\prime\prime}(s) (I_{(0,s]}(u) - s)(I_{(0,s]}(v) - s)(I_{(0,s]}(w) - s)\, dF^{-1}(s)$$...
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### Derivative of real functions including Re and Im

When deriving functions using Re, Im or Arg (and probably some other functions as well), ...
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When I use Limit to evaluate the limit \begin{align}\lim_{k \to 0} \frac{ (k+2)(\alpha^2 - \sqrt{\alpha^4 + k}) + k}{\alpha^2 - \sqrt{\alpha^4 + k} + 2 k}\end{... 12 votes 6 answers 10k 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 ... 12 votes 7 answers 3k views ### Trying to prove that x\sin(\frac{\pi}{x})\ge\pi \cos(\frac{\pi}{x}) for x\ge 1 As an intermediate step, 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 ... 9 votes 2 answers 1k views ### Bug in Integrate for Mathematica Bug introduced in 8.0.0 and fixed in 9.0.0 Consider the following: ... 15 votes 5 answers 1k 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 ... 8 votes 5 answers 743 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 ... 2 votes 1 answer 1k views ### Checking if first derivative is zero at a given point I have a function V[a_, h_, tau_] := (Sqrt/2) h (Sqrt a + h/3 tau)^2 now I want to check, whether its first derivative is zero in a given point. Say for ... 2 votes 1 answer 457 views ### Integral not simplifying I specify a function in terms of an integral and then try to evaluate it with Simplify. However, the answer is not really simplified to what it should be. ... 16 votes 5 answers 4k 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 ... 23 votes 2 answers 1k views ### How can the {x,y,z} points that fall on the outer boundary of a set of values be selected and smoothly surfaced? For a given set of x,y,z values, that may, or may not form a uniform shape, how can the center of the data cloud be found, and the surface points be located and a solid smooth surface created from ... 3 votes 1 answer 451 views ### n-fold symbolic integral in Mathematica I am trying to compute symbolically a n-fold integral (n is a parameter of a function) over, say, the cube [0,a]^n. My code looks like this ... 2 votes 1 answer 1k views ### Speeding up a slow indefinite integral I have found an alternative way to express an old solution: ... 3 votes 2 answers 3k views ### Plot Indefinite integral I'm having a hard time to find a solution to a numerical limitation on mathematica: ... 21 votes 2 answers 2k views ### Why does Mathematica give the wrong answer when integrating? Bug introduced in 8.0 or earlier and fixed in 9.0.0 I integrate Integrate[Exp[I Cos[b - c]] Cos[b], {b, 0, 2 Pi}] Mathematica gives: ... 13 votes 5 answers 21k 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: ... 15 votes 2 answers 17k 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 : ... 9 votes 3 answers 1k views ### Inverting a function in a certain region InverseFunction works well for globally invertible functions, like f = 2*# + 2 &; InverseFunction[f] ... 6 votes 1 answer 1k views ### Definite and Indefinite integral give different results for piecewise function I have the following function: f(q,y)= \begin{cases} \tfrac{11720+p}{37791360} & -11720<p<-7720 \\ 0 & \text{True} \end{cases} $$where p = 443\ y-777600\ \sin^{-1}\left(\frac{q \... 6 votes 3 answers 2k views ### recursive integration I am trying to do multiple integrations recursively. For instance, I would like to do the following equation for arbitrary integer n: \displaystyle R_n(t) = \int_0^t \mathrm dt' R_0(t-t') R_{n-1}(... 9 votes 1 answer 1k views ### Hankel Transform integrals won't work in Mathematica I'm trying to do this integral, which is shown on the Wikipedia page on the Hankel transformation:$$\int_0^{2\pi}\mathrm d\varphi\;e^{\mathrm im\varphi}e^{\mathrm ikr\cos(\varphi)} The answer is ... 5k views

### Using implicit differentiation to find a line that is tangent to a curve at a point

A few days ago I asked about using differentiation to find a line that is tangent to a curve at a given point. J.M. provided a very elegant way to solve these kinds of problems in Mathematica. Now I'...
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### 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.
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