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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4
votes
1answer
151 views

Difficulty with total derivative

I am trying to introduce myself to the total derivative Dt in the documentation tutorial/TotalDerivatives. But I ran into what I think is a problem. There seems to be a serious problem in the ...
4
votes
4answers
531 views

$L = \lim\limits_{z \to 1} \frac{1-z}{1-z^\ast}$ evaluates incorrectly

I'm about 90% certain this is a Mathematica issue, not me making a silly mistake. I'm trying to evaluate $$L = \lim_{z \to 1} \frac{1-z}{1-z^\ast}.$$ Naively entering ...
4
votes
2answers
285 views

How to force correct answers for Integrals of Cos[mx]*Cos[nx]? [duplicate]

This is a big problem if you do anything with a Fourier Series. This statement: Assuming[Element[{m, n}, Integers], Integrate[Cos[m*x]*Cos[n*x], {x, 0, 2 Pi}]] ...
4
votes
1answer
228 views

Efficient Dyson series implementation

I'm trying to implement a Dyson series \begin{array}{lcl} U(x,x_0) & = & 1 + \int_{x_0}^{x}{dy_1V(y_1)}+\int_{x_0}^x{dy_1\int_{x_0}^{y_1}{dy_2V(y_1)V(y_2)}}+\cdots \\ & &{} + \int_{x_0}...
4
votes
3answers
166 views

About the wrong evaluation of an integral

Here is an integral I've been studying in my research and I've just realized that Mathematica $8.0$ is unable to correctly compute it. I have 2 simple questions to ask: Is my code below correct? <...
4
votes
2answers
289 views

Calculate information entropy integral in infinite square well problem

In the context of information theory, entropy is a measure of uncertainty of a random variable. In quantum mechanics, the uncertainty principle states that $\Delta x\Delta k \ge 1/2$. The same can be ...
4
votes
2answers
389 views

How do I evaluate a symbolic integral involving Hermite polynomials?

I want to test a difficult integral : Integral on all reals of some complicated function involving the Hermitian polynomials, exponentials, squares, factorials, and being general considering any ...
4
votes
4answers
379 views

Differentiating space curves

I'm trying to do some very basic differential geometry of space curves. For example, a space curve $\gamma:\mathbb R\to\mathbb R^3$ has unit tangent and normal vectors given by $$t(s)=\frac{\gamma'(s)}...
4
votes
1answer
1k views

Computation of parametric integral

I am trying to compute the integral Integrate[(g^(u^(g - 1)))/(1 + u^g), {u, 0, t}] but as an answer I get my input expression. There must be something wrong ...
4
votes
1answer
1k views

Lie-Bracket of two vector fields

I'm new to Mathematica (installed couple of hours ago) and I need to compute a few Lie brackets between two vector fields $f$ and $g$. $$ f\left(\mathbf{x}\right) = \left( \begin{array}{c} ...
4
votes
1answer
120 views

Parametric numerical integration [closed]

I would like to solve the integral: $$ \iiint\frac{e^{-\sqrt{x^2 + k^2}}x^2 \sin{\theta}}{\sqrt{x^2 + k^2}} dx d\theta d\phi$$ defined in $x \in [0,+\infty]$, $\theta \in [0,\pi]$, $\phi \in [0,2 \pi]...
4
votes
2answers
92 views

Defining arbitary Derivatives of functions

If I have two functions like these f[a_,b_] and g[a_,b_]. Lets say ...
4
votes
1answer
120 views

Integration of a piecewise function

Define f[x_, i_] := Piecewise[{{1/n, 0 < x < (i - 1)/n}, {(i/n - x), (i - 1)/n < x < i/n} , {0, i/n < x}} ]; I would like to compute ...
4
votes
1answer
130 views

A 1D numerical integral Mathematica cannot compute, from physics

A well know result in theoretical physics is that a sum over Matsubara fermionic frequencies, i.e.: $$ S = \sum_{n=-\infty}^{\infty} h(\omega_n) \hspace{32pt} \omega_n=(2n+1)\frac{\pi}{\beta} $$ can ...
4
votes
1answer
146 views

Mathematica does not help in this integral

I am trying to solve this integral with assumptions: ...
4
votes
2answers
407 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: <...
4
votes
1answer
227 views

How to do this complex integration on the real line?

$m, r$ are parameters in the following integral: Integrate[z Exp[I z r]/Sqrt[z^2 + m^2], {z, -∞, ∞}] How to do this integration directly? The result should be <...
4
votes
2answers
130 views

Inconsistent results for equivalent converging symbolic integrals

I have looked at previous questions and I'm aware that this seems to be a known bug: Mathematica giving inconsistent results for symbolic integrals done in different ways. The origins for the bugs ...
4
votes
2answers
309 views

Limit won't compute

Does anyone know why the limit: ...
4
votes
1answer
220 views

Why does Assuming for Integrate not work as expected?

I'm trying to perform the following integral with Mathematica 7: ...
4
votes
1answer
226 views

How to make Mathematica express an equation in terms of previously defined functions

I was given a problem for my physics class, and perhaps I have taken it too far, but I wish to finish what I started. I have a function (i warn you that this does not look pretty) ...
4
votes
1answer
168 views
4
votes
1answer
114 views

Products of Differential Operators

I have a differential equation defined as the product of operators which I want to expand out into a polynomial in powers of $z\frac{d}{dz}$ $\qquad \prod_{n=1}^p(z\frac{d}{dz}+a_n)$ However when I ...
4
votes
1answer
185 views

Why can't the Bernstein Bears work normally?

Bug introduced in 7.0 and fixed in 9.0 I want to use the built-in BernsteinBasis[] to learn about Bezier curves. I tried the following code: ...
4
votes
1answer
112 views

How to apply RootLocusPlot correctly

I have to analyze the complex regions of variable c which is define obtained with InverseLaplace help in the form ...
4
votes
2answers
488 views

Using units and piecewise functions in Integrate

For simple cases, Quantities appear to be handled well by Integrate: ...
4
votes
1answer
1k views

Integral of Lorentzian yields different results depending on when parameter assigments are made

I'm evaluating the integral of a Lorentzian, which I know equals one. First I define the function and evaluate the integral in two slightly different ways. Surprisingly, I do not get the right answer ...
4
votes
1answer
587 views

Multi-dimensional integral in the complex plane with poles and essential singularity

I've passed the last week searching a way to numerically integrate this multi-dimensional integral in the complex plane at the poles and avoiding the singularity at z=0: $$ \oint_{C}\oint_{C\ auound\ ...
4
votes
2answers
351 views

Integral returns function of variable that was integrated over

Bug introduced in 8 and fixed in 10 When I enter ...
4
votes
1answer
151 views

Generalization of a rule to N arguments

I am trying to apply a series expansion on a function x[t1,t2,...tn], with an expansion parameter a. For ...
4
votes
2answers
151 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 ...
4
votes
1answer
142 views

How to collect terms with z-derivative?

my equations are very long (several pages). Here I will provide simple example: ...
4
votes
1answer
173 views

Bad performance of Integrate (and WolframAlpha) for an Integral of Bessel function of the first kind

The following returns unevaluated in WolframAlpha. Also in my machine Mathematica needs quite a lot of time to compute it. ...
4
votes
1answer
108 views

Formal Differentiation

I'm trying to formally differentiate an expression. (I am aware of how to differentiate formally?, but am unable to generalize the idea) For example: ...
4
votes
2answers
135 views

Imaginary terms in the derivative of Jacobi theta function (2) on the real line

I am trying to calculate/plot the derivative of the second Jacobi theta function $d\theta_2(0, e^{-\pi t} )/dt$. Calculating or plotting the function itself works fine: ...
4
votes
2answers
124 views

Adding assumptions changes the result of Integrate

When I try to calculate the following integral: ...
4
votes
1answer
312 views

Derivatives of functions with arbitrary number of variables?

I am trying to define a function f[x1,x2,...,xn] with n integer but not specified. And then I would like this funktion to behave ...
4
votes
0answers
69 views

Huge difference after changing a fraction to decimal

I have a limit to calculate. With[{p = 1/2, q = 0.1}, Limit[a^q Integrate[Sin[x]/x^p, {x, a, Infinity}], a -> Infinity]] gives correct result ...
4
votes
0answers
108 views

Exploring formal limit definition

I found a few demonstrations on the Wolfram Demonstration Project site that help users to explore the formal definition of a limit. That is, $$\lim_{x\to a}f(x)=L$$ if and only if for every $\epsilon&...
4
votes
0answers
119 views

Puzzled by Assumptions [duplicate]

I don't know if this has already been discussed. Integrate[BesselJ[2 m + 1, x], {x, 0, ∞}, Assumptions -> m ϵ Integers] ...
4
votes
0answers
112 views

Integral of DiracDelta giving an unusual answer

I have been getting a number of seemingly inconsistent solutions to integrals of Dirac delta functions in which the integrand evaluates to DiracDelta[0] at one of ...
4
votes
0answers
60 views

How to take the second derivative of a complicated function using limits [closed]

I have a very complicated function of one variable consisting of exponential and error functions combined in various ways. Mathematica can't take the derivative of this function in the usual way (...
4
votes
0answers
117 views

DiracDelta and version 10.0.0 [duplicate]

Note: This issue seems to affect version 10.0.0 only and is fixed in 10.0.1, 10.0.2, 10.1 and 10.2 When evaluating the following two inputs: ...
4
votes
0answers
90 views

Integrate yields complex value, while after variable transformation the result is real. Bug?

I have the follwoing integral: Integrate[1/Sqrt[0.7 + 0.3*(1 + z)^3], {z, 0, Infinity}, Assumptions -> z \[Element] Reals] >> -3.36354 - 3.85013 I the ...
4
votes
0answers
246 views

Strange behaviour of MMA in derivatives of some standard functions

There are some peculiar things to be discovered in derivatives of some standard functions in MMA: Strange behaviour Example 1: Abs We have ...
4
votes
0answers
142 views

Linearised Einstein equations

I need to compute the linearised Einstein Equations around a fixed metric $$g_{μν}=\text{Minkowski metric} + h_{μν}$$ which is not the flat metric. Does anyone know a Mathematica package or a ...
4
votes
0answers
328 views

Spherical harmonic derivative

Consider the following substitution Derivative[2, 0][S][th, ph] /. S -> Function[{th, ph}, SphericalHarmonicY[3, 0, th, ph]] which gives correct answer. While ...
4
votes
0answers
180 views

Version 8.0 integrates but Version 9.0.1 doesn't

I am trying to run the following integral in version 9.0, but it fails: ...
3
votes
6answers
2k views

Compute Triple Integral on spherical coords

I need to compute: $\int \int \int z dxdydz$ over the domain: $\{x^2+y^2+z^2\leqslant 16,z\geqslant 0\}$ Im trying to use spherical coords as: $$\int_{0}^{2\pi} \int_{0}^{\frac{\pi}{2}} \int_{0}^...
3
votes
3answers
191 views

Plotting the region bounded by parallel lines

I am working on the integral $$\int\int_D (2x+y)^2 e^{x-y}dA,$$ where $D$ is the region bounded by $2x+y=1$, $2x+y=4$, $x-y=-1$, and $x-y=1$. I need to shade the region bounded by these lines, so I ...