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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0
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0answers
20 views

different values of integral with/without bogus assumptions

I do not understand why I get different answers for the following two identical integrals. ...
3
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1answer
151 views

Fredholm integral equation of the second kind with kernel containing Bessel and Struve functions

I need to solve this Fredholm integral equation of the second kind: f[s]+integrate[f[t] K[s,t],{t,0,1}]=s where ...
17
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5answers
5k views

How to find the domain and range of a function with Mathematica?

I'm studying calculus and in some exercises I am asked to find the domain and range of a function. Does Mathematica have already a built-in function for this? I can imagine some ways of doing so, ...
5
votes
1answer
209 views

Prove an identity in quantum harmonic oscillator

Problem: In the context of quantum harmonic oscillator the eigenfunctions are given by: $ u_n(x) = (N_n/\sqrt{b}) H_n(x/b) \exp\left[-x^2/(2b^2)\right] $, where $N_n$ is the normalization factor: $ ...
5
votes
2answers
278 views

How can I calculate the limit without using the L'Hopital's rule

I need to prove this limit without using the L'Hopital's rule: $$\lim_{x\to 0} \frac{(1+a\,x)^{1/4} - (1+b\,x)^{1/4}}{x} = \frac{a-b}{4}$$ How can I do it in Mathematica?
2
votes
3answers
261 views

Need help evaluating definite integral to a function of Y

Suppose $Y = \sqrt{2T}\cos(U)$, $ 0 \le u \le \pi $, and $ 0 \le \cos^{-1}(\frac{y}{\sqrt {2t}}) \le \pi ) $, so $ -1 \le \frac{y}{\sqrt{2t}} \le 1 $, with all $ \mathbb{R}$. The iterated integral ...
1
vote
1answer
125 views

Under what conditions does Mathematica give a natural antiderivative when integrating?

Natural antiderivative is defined as follows, using Fourier transform: $$f^{(-1)}(x)=\frac{i}{2\pi}\int_{-\infty}^{+\infty} \frac{e^{- i \omega x}}{\omega} \int_{-\infty}^{+\infty}f(t)e^{i\omega t}dt ...
3
votes
2answers
255 views

Integral converging in M9 but not in M10

Background: Mac OSX 10.9.4, Mathematica 9.0.1 vs. Mathematica 10.0 (both versions are Student Editions). I have a notebook. It used to evaluate fine in Mathematica 9. I upgraded to 10 and, without ...
25
votes
4answers
3k views

How to find the period of an arbitrary mathematical function?

Is there a function to find the period of an arbitrary (possibly complex) function in Mathematica?
0
votes
2answers
250 views

Integrating polynomial functions over polytopes with an add-on package

There is a Mathematica package to evaluate integrals over polytopes: http://library.wolfram.com/infocenter/Books/3652/ In the documentation (Functions.nb file) I ...
6
votes
4answers
766 views

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 ...
1
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2answers
634 views

Computing 10-dimensional volume of a 9-sphere [closed]

I'm trying to compute 10-dimensional volume of a 9-sphere with radius r using Monte Carlo. ...
13
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5answers
294 views

Mismatch between numerical and analytic evaluation of an integral

I evaluated the following integral NIntegrate[Sqrt[r] Abs[Cos[(k + 1/2) Pi r]], {r, 0, 1}] getting as a result 0.413232 for ...
2
votes
0answers
90 views

Result of symbolic integration changed drastically by making assumptions

I would like to know the underlying reason for different outcomes for the two integration operation below. One of them includes a few assumptions, otherwise both have the same integrand: ...
31
votes
5answers
2k views

Generating evenly spaced points on a curve

In the KnotData package a simple command such as points = Table[KnotData[{3, 1}, "SpaceCurve"][t], {t, 0, 2 Pi, 0.1}]; will ...
3
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0answers
86 views

Derivative of generating function (Example from documentation)

In the documentation for GeneratingFunction, the following example is given under Examples -> Properties & Relations -> Derivative: ...
0
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0answers
50 views

Getting long complex-valued integrals when simpler real-valued expressions exist

I have a long list of real-valued functions I'd like to integrate symbolically. For many of them, Mathematica gives me results with long complex-valued expressions involving weird functions such as ...
20
votes
3answers
1k views

Can we teach Mathematica about functional differentiation?

The key relation for functional differentiation is $$\frac{\delta}{\delta f(y)}f(x)=\delta(x-y), $$ where $\delta(x-y)$ is the Dirac delta function, and the usual properties of differentiation (e.g. ...
1
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1answer
130 views

Is it necessary to introduce ImplicitRegion and ParametricRegion in version 10?

In version 10.0, Mathematica introduced a new function about region plotting: ImplicitRegion. But I'm wondered that haven't this function been already existed in the older version? That is, ...
0
votes
1answer
42 views

Evaluation of integral [closed]

I want to evaluate an indefinite integral using the Integrate command, and I know that the answer can be written in terms of elementary functions (square roots, logs etc.) but Mathematica seems to ...
5
votes
0answers
89 views

Convoluting inverse square root with Gaussian

I would like to convolute the inverse square root on the interval [0,inf] with a Gaussian function, like so: ...
5
votes
3answers
202 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 ...
0
votes
1answer
69 views

Integral calculation in Mathematica [closed]

I need to calculate this integral using Mathematica: ∫ t^z * ln(t) dt from x to y with the assumptions that z is an integer and 0 < x < y This is what I'm trying to do: ...
3
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0answers
62 views

Symbolic integration of elliptic functions

Is there some clever way to integrate products of elliptic functions $\wp(z;g_2,g_3)$ or $\zeta(z;g_2,g_3)$ in Mathematica? Mathematica seems to be able to integrate functions like ...
8
votes
1answer
200 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}]
2
votes
2answers
97 views

Integrate returns unexpected result

Consider the following function $$g(x,y):= \frac{1}{( (1+y)^2+x^2 )( 1+ax^2y^2 )^2}$$, where I assume that $y\geq 0$ and $a\in (0,1]$ is a parameter. When I try to evaluate the integral $\int ...
4
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0answers
62 views

How to save the result in the Notebook (.nb) and shut down the computer when the calculation is done

I have a notebook(.nb) which its calculation is time-consuming and long. I can not observe it if the calculation is done or not. Therefore, I want to : (1) Save the results in the notebook(.nb) ( I ...
4
votes
2answers
100 views

Integrating a periodic function

I have a periodic function ff: ff := Function[x, Piecewise[{{ff[x - 1], x >= 1}, {2 x, 0 <= x < 1}, {ff[x + 1], x < 0}}]] Plotting it works fine: ...
0
votes
2answers
128 views

Integral with unreliable result

I want to calculate $\int_R^1 \sqrt{r} |\cos((k+\frac{1}{2})\pi r)|dr $ and I get a result from Mathematica. Then I try to check the result putting the value of $k$ and $R$, (k=1 and R=0.5) in the ...
15
votes
4answers
6k views

Finding unit tangent, normal, and binormal vectors for a given r(t)

For my Calc III class, I need to find $T(t), N(t)$, and $B(t)$ for $t=1, 2$, and $-1$, given $r(t)=\{t,t^2,t^3\}$. I've got Mathematica, but I've never used it before and I'm not sure how to coerce ...
8
votes
2answers
577 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 ...
6
votes
1answer
5k 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 ...
1
vote
2answers
124 views

Higher derivative [closed]

Is there more efficient way to rewrite this code in order to compute 2nd and higher derivatives of r=sqr{x^2+y^2+(z-a)^2}? ...
0
votes
1answer
91 views

Integrate and NIntegrate yield different results for double integral

Evaluating a double integral with bivariate normal distribution yileds widely different results depending on the method used. I define a bivariate normal distribution with {10, 3} and {8, 1.5} as ...
2
votes
1answer
86 views
1
vote
1answer
131 views

Check for holomorphy of a function

Given a (rather complicated) function H(z), what is the best approach to check symbolically whether it is holomorphic? What I tried is checking explicitly the ...
6
votes
4answers
231 views
1
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0answers
97 views

Symbolic matrix calculus: What's new in Version 9

I have seen some of the related posts, which ask about doing matrix calculation on tensors unknown dimensions. One of the posts mentioned that version 9 has some capability, but it was concerned about ...
10
votes
6answers
4k 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 ...
3
votes
0answers
70 views

Using Mathematica to help obtain correct analytic formula for logarithm integration

I need to program into my Mathematica code the analytic form for the result of the integral: $$I(a,b)=\int ^1 _0 dx \frac{\ln(x-a)}{x-b}$$ that is valid for all complex $a$ and $b$ (but $\text{Im } ...
3
votes
3answers
672 views

How to integrate a function over a 3D planar polygon?

I am trying to integrate a function over a planar polygon in 3D. In 2D, this is fairly straightforward, using either answer from this question (I use the second answer). If we use an equilateral ...
8
votes
1answer
105 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 ...
2
votes
1answer
172 views

Mathematica policy for correctness of results [closed]

Does Mathematica provide any kind of warranty that their calculations are correct? Say I'm running a billion dollar company, and I relied on Mathematica to do calculations, like aerodynamics or car ...
18
votes
2answers
276 views

Negative probability?

I am trying to get the sum of the squares of seven random variables, all uniformly distributed. This is what I tried. ...
0
votes
1answer
75 views

A problem with “RiccatiSolve::ncsol: Cannot compute a stabilizing solution to the matrix equation ”

I have a SSM ( State Space Model ) and I want to calculate its optimum gains unisng LQOutputRegulatorGains[] command. When I run the code, it solves the equations ...
0
votes
0answers
92 views

Complex Convolution

I am attempting to integrate a convolution variable using the following code. However, the program is taking too long to complete the integration. Does anybody have any coding tips that may make ...
0
votes
1answer
79 views

Integral over geometric region [duplicate]

I'd like to calculate this integral $$ \int_E y\ dydz $$ where $E = \{ (x,y,z) \in R^3 : z^2+6 < y^2 < 5z \}$ By hand i've got $\frac{1}{12}$ but i'm not sure, and i'd like to verify this ...
2
votes
2answers
152 views

Finding Expectation of function of a Log-normal distribution

Say $Y=g(X)$ and $p_X = \frac{e^{-\frac{(\mu -\log (x))^2}{2 \sigma ^2}}}{\sqrt{2 \pi } x \sigma }$ is Log-normal density function: [Wiki] Find E[Y]? Since $E[Y] = \int_0^\infty y f_Y \ dy = ...
0
votes
1answer
82 views

Defining a function that differentiates another function with rule replacement

This question is related to the question I posted here. In that question, I thought I have simplified my original problem into an equivalent concise version, but I found that it is not. So I decided ...