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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0answers
99 views

Interpretation of output derivative expression [duplicate]

I am trying to write Mathematica file to find Div, Grad, Laplacian, etc. in coordinates ...
4
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1answer
1k views

finding poles of a function

Is there a command to find the poles of a function $f=f(z)$? example: let $$f(z) = \frac{1}{z^2-1}$$ then we know that the poles are at $z=\pm 1$ but is there a special command in mathematica to do ...
1
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2answers
269 views

Finding normals to a curve through a given point

I want to find x on curve $y(x)$, where curve $y(x)$ is perpendicular to point $(x_0,y_0)$. for this I minimize the distance between points $(x,y(x))$ and $(x_0,y_0)$, here's an example for point ...
2
votes
1answer
1k views

Laurent series expansion

Can someone share how to find a Laurent series expansion of $$f(z)=\frac{1}{(z^2-1)(z^2-4)}$$ centered at zero on the annular disk $1<|z|<2$?
7
votes
1answer
403 views

Numerical contour integrations in the complex plane - contour deformation gives different answer for analytic kernel

I am trying to do a contour integration in Mathematica numerically. In particular, I'm checking the identity: $$ H_m^{(1)}(z) =\frac{i^{-m}}{\pi}\int_{-\pi/2 + i \infty}^{\pi/2 - i \infty} \exp[i m ...
2
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0answers
152 views
3
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1answer
281 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 ...
0
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1answer
231 views

Can I take derivative of a function that represents a piece of music?

Can I take the derivative of a function that represents a piece of music? The music is given in .mp3, .midi or whatever comes in handy.
13
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2answers
799 views

Numerical partial derivative

For a one-variable numerical function, it's simple to calculate the derivative at a point with Derivative as Szabolcs has pointed out before: ...
0
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1answer
446 views

No output for N[Limit[Integrate[Sin[x]^n/(Sin[x]^n + Cos[x]^n), {x, 0, Pi/2}], n -> Infinity]]

Is my input below correct? I've received no response from Mathematica. N[Limit[Integrate[Sin[x]^n/(Sin[x]^n + Cos[x]^n), {x, 0, Pi/2}], n -> Infinity]] Thank ...
2
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0answers
48 views

Verifying integration by parts of an arbitrary function [duplicate]

Suppose that I have an arbitrary differentiable function $G$ which depends on $s$: $G(s)$. Now suppose that I want to do the following integral: $$\int ds \;\; s \frac{\partial G(s)}{\partial s}.$$ ...
1
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1answer
183 views

Derivative inside a function

I am trying a problem that should be simple: to create a function that contains a derivative. I need to do a more complex example, but even with the simplest I'm having trouble: Let's consider ...
1
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0answers
360 views

Cauchy principal value integral of a list of numbers. How?

I have a list of numbers that are numerical samples of a function for which I need to find the Cauchy principal value integral. I thought I should be able to combine Interpolation with Integrate to do ...
9
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1answer
364 views
5
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3answers
762 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') ...
13
votes
2answers
353 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 ...
3
votes
1answer
607 views

Evaluating double Integral

While trying to evaluate the integral $\int_{y=0}^{x_2}\int_{x=0}^{min(x_1,y)} n (n - 1) (1 - y)^{(n - 2)}dxdy $ , Mathematica does not seem to yield any results. ...
6
votes
4answers
811 views

How do I obtain the correct double limit?

The command Limit[(Sin[x^2] + Sin[y^2])/ (x - y) /. x -> 0, y -> 0] (* 0 *) I think that Mathematica finds the iterated limit instead of the double ...
16
votes
4answers
6k views

Finding Limits in several variables

Is there a way to find a limit of a multivariable function, like $$\lim_{(x,y)\to (0,0)} f(x,y)$$ with Mathematica? When $f$ is continuous, we can use $$\lim_{(x,y)\to (0,0)} ...
6
votes
1answer
907 views

How to expand a function into a power series with negative powers?

Is there any way to expand this expression a+b(1-Exp[-T/(b c)]/(z-Exp[-T/ (b c)]) (where a, ...
2
votes
2answers
369 views

Write a function that returns the logarithmic derivative

How can we write a function that if we input an expression f, it returns the log derivative $\frac{1}{f} \frac{df}{dx}$. We have to use a conditional or pattern test so that the function accepts any ...
4
votes
1answer
652 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 ...
2
votes
1answer
150 views

Integration of a rational function

I am trying to solve the following rational integral (I got assistance here): $$\int_{-\infty}^{\infty}{\frac{a+x}{b^2 + (a+x)^2}\frac{1}{1+c(a-x)^2}}dx$$ where $\{a, b, c\}\in \mathbb{R}$. I would ...
3
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0answers
144 views

Is this result wrong because of calculation time? (and more questions about Assumptions)

I am confused with the Integrate Given by Mathematica. First let's see a one-dimensional case: ...
0
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1answer
226 views

Integrate of Heaviside Function returns an error answer

I try to evaluate an Integral of a Heaviside function, but it turns out with an erroneous output(zero). Here is my integral: ...
1
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3answers
624 views

How do I evaluate several n-th derivatives of a function at one point?

I have a question where I have to compute a Table containing $f^{(n)}(0)$ for n = 1, ..., 5, where $f^{(n)}$ denotes the $n$th ...
9
votes
2answers
608 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
votes
1answer
289 views
13
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1answer
288 views
4
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2answers
136 views

Strange behavior of Integrate

I'm trying to calculate the following integral using Mathematica 9.0.1.0 a11=Integrate[Abs[Sin[b+x]],{x,0,2*\[Pi]}] This should be a simple problem; however, it ...
1
vote
1answer
277 views

how to reduce the time to calculate triple integral

i want to evaluate the following integral, but it takes 15 min to do it. i must reduce the time to calculate it, i have to do a table and create a graph, but takes too long. anyone have an idea ? ...
1
vote
1answer
133 views

Assumptions are not being applied to integral

I am trying to use assumptions to simplify an integral, but I can't seem to get the assumptions to apply themselves. Here is the code I am using: Note the expressions in the assumption can be found ...
0
votes
1answer
69 views

Is there a more idiomatic way to solve this implicit differentiation problem?

Here's an exercise from a calculus text by Larson and Edwards: Find the rate of change of the distance between the origin and a moving point on the graph of $y = x^2 + 1$ if $dx/dt = 2$ ...
2
votes
0answers
250 views

Derivatives of list elements

Could someone explain the odd behavior of the Derivative function when drawing arguments from lists? We have, ...
0
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0answers
34 views

Plot derivative of integral with changing limits [duplicate]

My code is: Plot[D[Integrate[E^(t*(-x^2)),{x,t+2,t^2}],t],{t,-2,3}]. The errors/warnings are: ...
4
votes
3answers
1k views

Definite integral takes a very long time

I'm trying to integrate the following function: $$ r_e=\int_0^z\frac{dz}{\sqrt{\Omega_m(1+z)^3+\Omega_\Lambda}} $$ where $\Omega_\Lambda=1-\Omega_M$ and both $\Omega_\Lambda$ and $\Omega_M$ are real ...
5
votes
2answers
2k 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 ...
0
votes
1answer
115 views

Simple Question: Derivative of a path

I have a path V[t_]:={3t^2,2,6t}, and I define F[t_]:=V[t]/Norm[V[t]]. I simply want the derivative of ...
2
votes
2answers
263 views

integrating slightly complicated expression which includes Abs

I have been waiting for Mathematica to give me something for the following integral, errors welcome, but it has been "running" for almost 30 minutes now. ...
7
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2answers
1k views

How to change coordinates of integration?

Is there some built-in routine, some easier method to change variables of integration, without/before solving the integral? Say I have ...
3
votes
3answers
295 views

Proving a recurrence in Mathematica

I have $$j_n=\int_0^1 x^{2n} \sin(\pi x)dx.$$ How do I show that $$j_{n+1}= \frac{1}{\pi^2}(\pi- (2n+1)(2n+2)j_n)\, ?$$ I keep getting a recurring integration by parts and I can't simplify it. ...
5
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2answers
176 views
4
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0answers
191 views

Strange result from evaluating a Limit expression

When I evaluate the limit of (1-Cos[x]) Sin[1/x] at x = 0 in Mathematica 9, I obtained ...
5
votes
2answers
300 views

Mathematica cannot calculate a limit

When I evaluate Limit[E^(-n)*Sum[n^k/(k!),{k,0,n}], n -> ∞] Mathematica gives me the result ...
8
votes
2answers
525 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 ...
-3
votes
1answer
159 views

Limit of a subsequence

I am trying to calculate limit for a subsequence <${ a }_{ {n }^{ 2} }$> because I was trying to see that limits of sequences and subsequences are both same. I was able to generate the sum for ...
1
vote
0answers
120 views

Getting poles of a Gamma functions [duplicate]

Why do the following 2 sequences give different answers? n = 1.5 Series[Gamma[0.5 - n - x], {x, 0, 2}] Series[Gamma[-1 - x], {x, 0, 2}] (..clearly the output from the second expression is ...
1
vote
2answers
368 views

Implicit differentiation to find the slope of the tangent line to the graph at the indicated point [duplicate]

I used ContourPlot to plot the graph of x Sin[Pi y] - y Cos[Pi x] == 1 with the following input: ...
16
votes
1answer
442 views

Speeding up trigonometric integral

Context On a possible non trivial toric topology for the Universe (nothing less!). Problem I would like to carry out the following integral for $\ell=2,4\cdots 20$. $$\int _0^{\pi }\int _0^{2 \pi ...