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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3
votes
1answer
981 views

Problem with limit that requires L'Hôpital's rule to compute

Consider the following limit. Limit[(a - Sqrt[a^2 + x])/(a^2 - a*Sqrt[a^2 - x]), x -> 0, Assumptions -> {a > 0}] Mathematica 9.0.1.0 gives ...
1
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1answer
292 views

Is it possible to give the closed-form of the stiffness matrix of triangular prism element?

Suppose the vertices of triangular prism are 1, 2, 3 (bottom triangle), 4, 5, 6 (top triangle), the coordinates of the vertices are $$ (x_i, y_i, z_i), i=1,2,3,4,5,6. $$ We can write the stiffness ...
1
vote
1answer
780 views

Numerical saddle point problem of a function of many variables

I want to find the following saddle point of the function of 10 variables: . I am not able to find the stationary points by setting the first derivatives to 0, so I need do optimize the function ...
11
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2answers
998 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 ...
0
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1answer
87 views

One more question about Assumptions and Integrate

There are many questions about assumptions here. Many of them are problem specific. That is why I want to ask a question in the most general way. Why does Mathematica solves this ...
0
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1answer
194 views

Integration of DiracDelta

I just came across the following curiosity, and I'm not sure whether it's a bug or just me missing something. Here's a minimal example: I defined the following "function" ...
13
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2answers
526 views

How to compute the residue of $e^{z-\frac{1}{z}}$ at z=0?

I've tried the following but it didn't work: Residue[Exp[z - 1/z], {z, 0}] not even this: Residue[Exp[1/z], {z, 0}] ...
5
votes
3answers
448 views

Creating a function with integral zeroes of the 0th, 1st, and 2nd derivatives

I would like to be able to randomly generate functions, each of which satisfies $f : [-10, 10] \rightarrow [-10, 10]$ All the zeroes, critical points, and inflection points have an integral $x$-...
1
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1answer
1k views

Error Function Integral (Erf)

Any idea how to solve analytically this integral Integrate[(a Erf[a Sqrt[b/(a^2 + b)] c])/(a^2 + b)^(3/2), a] I tried substitution u=a^2 + b, but it didn't work. ...
-8
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1answer
194 views

Alternative closed forms for digamma expressions [closed]

What commands should I use such that Mathematica / W|A express $$\psi(1+i)+\psi(1-i)$$ exclusively in terms of trigonometric functions?
1
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2answers
113 views

Unknown limit of an array of area integrals

Could someone explain why Mathematica can't finish computing this limit (this is a limit of an array, when n -> Infinity (n ...
12
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2answers
1k views

Why do I get a different value when I change the order of integration?

I think the following two-dimensional integrals should be equal, since they both integrate the function over the half plane defined by $t>\tau$. $$\int_{-\infty}^\infty \mathrm{d}t \int_{-\infty}^...
1
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1answer
150 views

Improper mutiple integral

I would like to solve the following integral: ...
6
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1answer
683 views

Using Boole to integrate over region

I have a complex and long piecewise function to integrate, so I have written a Mathematica script that cuts the whole range of integration, evaluates the integrand in each region and then integrates ...
9
votes
2answers
1k views

Double integral over a parallelogram

Problem: Evaluate the following double integral $$\iint_D (14 x^2+61 xy+42 y^2)^{3} \, dxdy$$ where $D$ is a parallelogram between these four lines: $2x+7y=6$, $2x+7 y=-6$, $7x+6 y=6$ and $7x+6y=-6$....
3
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1answer
642 views

A Bessel & Struve functions related integral

I try to numerically compute this integral and I don't figure out why on earth Mathematica is not able to do it. Is my input correct? Does it possibly have a closed form? ...
0
votes
3answers
220 views

How to remove the 1. when differentiating 0.5x^2

My question is very basic, When using the Differentiate functionality of Mathematica in this way: D[0.5 x^2, x] This results in: ...
2
votes
2answers
328 views

Another fractional part related question

Having known the fact that the evaluation of the integrals containing fractional parts often contains errors, I wonder if the integral below possibly evaluates to $0.7$ or this is just a coincidence ...
1
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1answer
1k views

Calculating electrostatic potential from arbitrary charge density

In electrostatics, one often desires to know the electrostatic potential generated from the outside of a charge distribution; however, I am interested in finding the electrostatic potential inside of ...
1
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1answer
264 views

Help with finding solution for utility function [closed]

I'm brand new to Mathematica and having trouble finding a solution for Lh based on the first order condition. I'm not sure if it is something I'm doing in Mathematica or something wrong with the ...
0
votes
1answer
474 views

Revolving the area between two functions about an axis

I'm trying to graph the revolution of the area between two functions about an axis, but I'm new to Mathematica, and not sure how to proceed. My area is defined by ...
6
votes
2answers
597 views

Using units and piecewise functions in Integrate

For simple cases, Quantities appear to be handled well by Integrate: ...
11
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2answers
3k views

Replace rule with function? Derivatives don't evaluate

Say I have an expression (call it expr) involving a function, f[x]. I'd like to be able to evaluate that for a particular choice of ...
0
votes
1answer
80 views

Showing result of derivative and other functions [closed]

I have a problem with my Mathematica 7.0 When I write: Derivative[x^2, x] Mathematica returns as result : Derivative[x^2, x] ...
1
vote
2answers
406 views

NDSolve boundary query / Extracting values from solution

I have a function $O_{2}$ with boundary conditions $O_{2}(r_{o}) = p_{o} $ and $O_{2}(g) = O_{2}'(g) = 0$. I plot it using the solver code below; ...
0
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1answer
71 views

Different integral when used with assumptions

I want to calculate an integral, but Mathematica gives me different results if I am more specific with the Assumptions option of ...
9
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2answers
444 views

An integral with a fractional part in 3 dimensions

The evaluation with Maple suggests the triple integral is around $1$, but Mathematica tells it's $0.0958758$. When using the code ...
2
votes
1answer
182 views

Need to take infinite sum of residues, is there a way to choose the order of operations for ReleaseHold?

I'm computing an infinite sum of residues. I want to do something like this: ...
8
votes
2answers
368 views

Working with a system of differential equations that cannot be solved explicitly

I have to work a lot with three functions $\;o_1(t), o_2(t), o_3(t)\;$ that are solutions to a certain system of differential equations: ...
1
vote
1answer
733 views

Integral over squared Hermite polynomial

I would like to calculate the uncertainty of the nth Eigenstate of a 1-dim harmonic oscillator. To obtain the result I have to solve the integral $$\int_{-\infty}^{\infty} \psi^* x^2 \psi \:dx$$ ...
5
votes
3answers
454 views

Complex limit gives wrong answer

I'm evaluating $$\lim_{z\to e^{i \pi/2 n}} \frac{z-e^{i \pi/2 n}}{z^{2 n}+1}$$ with Mathematica 9.0.1: ...
3
votes
2answers
500 views

Incorrect value of infinite sum

Wolfram Alpha and Mathematica give an incorrect result (numerically) for the following infinite sum: N[Sum[((-1)^n)/(n - (-1)^n), {n, 1, Infinity}]] (* 0.700059 *) ...
2
votes
2answers
358 views

Triple series - evaluation delayed

Trying to figure out if the infinite triple series has a nice closed form. It seems Mathematica is unable to help us here. Numerically, things remain the same, no response. Could you help? ...
12
votes
4answers
7k 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 ...
4
votes
2answers
531 views

Limit won't compute

Does anyone know why the limit: ...
1
vote
1answer
121 views

Asymptotics of sum of binomials

I would like to find the asymptotics of Sum[Binomial[n,i]*1/i^((n+1)/2),{i,1,n}] I saw Find asymptotics of $\sum\limits_{i=0}^{n/3} 2^i \binom{n-i-1}{\frac{2n}{...
15
votes
6answers
696 views

No response to an infinite limit

Trying to compute this limit, things seem to get frozen, even for hours. What would you recommend to fastly compute it? ...
1
vote
1answer
4k views

How to find the area between two curves

Find the area between y = 1 - x/π and y = Sin[x]. Here is what I have typed in so far into Mathematica: ...
10
votes
2answers
811 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 ...
2
votes
1answer
204 views

How can I get the limit of a certain infinite product?

A first look at it suggests the fact the limit is precisely $1$. To check that I tried Mathematica, but no output so far. Most probably it converges very slowly. Is there any way to get the limit? <...
8
votes
1answer
449 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: $ ...
2
votes
1answer
223 views

Function derivatives in arbitrary dimension

I have a higher dimensional function: $$f(x,t)=-\ln(t)-\ln(1-\left\|x\right\|^2-t^2)$$ where $x\in\mathbb{R}^n,~t\in\mathbb{R}$. I want to compute the derivatives with the commands ...
0
votes
1answer
947 views

Does Mathematica know if an integral is convergent or not?

Consider the following integrals that I asked Mathematica to do, ...
12
votes
3answers
2k 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 ...
11
votes
3answers
980 views

Pulling constants out of integrations [duplicate]

The following is the code and output of a Mathematica command How do I get Mathematica to remove $g(y)$ outside the integral?
0
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1answer
88 views

Having Trouble Solving this System of ODEs

I am having trouble finding any solution to this ODE: ...
6
votes
1answer
149 views

Is it possible to find a function from first few terms in the expansion

Is it possible to find a function if first few terms of the expansion is known. For example if I have this series $f(x)=\frac{k^3 x^2}{6}-\frac{k^5 x^4}{120}+\frac{k^7 x^6}{5040}-\frac{k^9 x^8}{...
11
votes
3answers
337 views

Integrating a BesselJ integrand to obtain the same result as Maple 16

I would like to check the following integration: Integrate[y Integrate[1/x BesselJ[1, x Exp[I π/4]] BesselJ[1, x Exp[-I π/4]], {x, 0, y}], {y, 0, r}] ...
4
votes
2answers
484 views

Can Mathematica (or its extensions) do integration following Risch algorithm?

I wonder whether there are option for indefinite integration in Mathematica that alow to choose the algorithm? Is there an option to use this algorithm in Mathematica?
4
votes
1answer
296 views

Volume within parameter space

Imagine a parameter space with variable 0<p<1, 0<e1<1/2 and 0<e2<1/2. ...