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
2answers
148 views

Well-defined symbolic integral leading to ConditionalExpression

I would like to determine a closed-form expression for the following symbolic integral $$ \int_{-1/2}^{1/2} \!\!\!\! \mathrm{d} x \int_{-1/2}^{1/2} \!\!\!\! \mathrm{d} y \, \frac{1 + b x + c y}{1 + e ...
3
votes
3answers
75 views

Multiple integrals where the number of integrals is aribtrary

I know that I can do, say, a triple integral such as $$\int_{a_x}^{b_x}\int_{a_y}^{b_y}\int_{a_z}^{b_z}f(x,y,z)\,dz\,dy\,dx$$ with the input ...
-5
votes
1answer
79 views

Show steps to compute limit [closed]

I'm trying to use Mathematica to compute a limit, because I have no idea how to compute it by myself. The limit should be: $$\lim _{n\rightarrow \infty }\dfrac {2^{n+1}-n-2}{2^{n}}$$ Any ideas?
0
votes
2answers
252 views

Multidimensional Numeric Integration

I can't manage to calculate numerically the following multidimensional integral: $I(X,Y,T)=\int_{0}^{T}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(...
0
votes
1answer
211 views

Cauchy Principal Value integral- no result is obtained [closed]

I have a particular Cauchy Principal Value integral that I need to numerically solve for my thesis research. It is the following $$ \int_{-1}^{1}dx\frac{\mathbf{p}^{2}+\mathbf{k}^2-2|\mathbf{p}||\...
4
votes
2answers
267 views

Using Mathematica to confirm Bernoulli's inequality

I have several challenges that I want to confirm is true. I have chosen this one because it is rather simple (proof by induction). There are times when I do not want to spend ages trying find proofs. ...
0
votes
0answers
27 views

Use implicit differentiation to find the slope of the tangent line to the curve 4 xy^3+ 5 xy= 45 at the point ( 5 , 1 ) [duplicate]

Use implicit differentiation to find the slope of the tangent line to the curve 4 xy^3+ 5 xy= 45 at the point ( 5 , 1 )
1
vote
2answers
154 views

Is there a way to do a cyclic convolution analytically in Mathematica?

Let's say I want to convolve a Gaussian function Pulse[t_] = Exp[-(t - Mid)^2/(2 sig^2)]; with an exponential broadening function ...
6
votes
2answers
470 views

Convolve does not get the correct answer

Convolve[Sinc[x], Exp[-x^2], x, X] (* E^-X^2 π *) is obviously false, but why? Any suggestions ?
3
votes
3answers
131 views

How to Integrate trivial products of DiracDelta

A long while ago I was able to integrate with Mathematica: $$\int_0^1 \delta(1-x)\delta(x) f(x) \,dx = 0$$ using ...
0
votes
1answer
136 views

How to do this triple integral?

I have a triple integral which is kind of complex, and I want to use Mathematics to help me do the integral. However, when I press "Enter" and "Shift" the software get stuck. I wonder whether this ...
0
votes
2answers
279 views

Finding the equation of tangent lines where a function has a specified slope [closed]

A function is defined as f[x_] := 3x^4 + 8x^3 - 24x^2 - 48x + 19 I need to find the equation of the tangent lines at the points where the tangent line has a ...
2
votes
5answers
354 views

Simplify an integral involving trigonometric functions

We were doing variation of parameters in differential equations tonight and had to do the following integral, the result given by hand calculations. $$\int \cos t\tan^2 t\,dt=\ln|\sec t+\tan t|-\sin ...
1
vote
1answer
78 views

Integrating with assumptions and limits of integration [duplicate]

I have an integration with assumptions and limits of integration. I tried to use Mathematica to solve this problem, but I cannot get any results. ...
4
votes
2answers
574 views

Real integral giving imaginary answer

Hello I am trying to evaluate the following integral Integrate[4279/Sqrt[0.6817 + 0.3183*(1 + x)^3], {x, 0, 20}] my mathematica 9 gives me ...
-1
votes
1answer
92 views

how can use $\mathop {\lim }\limits_{x \to 0} \frac{{e^x - e^{\sin \left( x \right)} }}{{x - \sin \left( x \right)}}$ [closed]

How can calcul $$\mathop {\lim }\limits_{x \to 0} \frac{{e^x - e^{\sin \left( x \right)} }}{{x - \sin \left( x \right)}}$$
1
vote
0answers
757 views

Mathematica can't, or won't evaluate an integral (again)

I'm relatively new to Mathematica and really need some help here to spot what's going wrong with my integration issues; Mathematica doesn't evaluate my integral, after a fair amount of time running, ...
0
votes
1answer
268 views

How to make Mathematica show intermediate steps in integral [duplicate]

I want to know how Mathematica evaluates this integral: $PV \int _ {0} ^ {i \infty} d \tau \, \frac{e^{-\frac{\tau^2}{2 M^2}}(b^2 - 3 τ^2)^2 (|b^2+\tau^2|-(b^2-\tau^2))}{\tau (\tau^2 - b^2)}$, where ...
0
votes
1answer
263 views

Solving Integro-Differential equations

I need help solving this equation. Is there a built in function that solves this type of equations? DSolve wouldn't work. Updated equation: ...
0
votes
2answers
297 views

Why does FindMinimum return 'The function value Null is not a real number'?

I am working with the following function ...
4
votes
1answer
275 views

Series approximation to integral

I would like to approximate the integral $$ \int_0^\infty dy\,\frac{1}{\sqrt{2\pi y\sigma^2}}\exp\left(-\frac{(x-y)^2}{2y\sigma^2}\right)f(y), $$ as a series expansion in the limit $\sigma\rightarrow ...
2
votes
1answer
126 views

Calculate exact Probability of product of normal random variables

Find the value of $P[\Pi_{i=1}^{10}X_i > C]$ for $C=5$, where $X_{10\times 1}$ is a random vector with $10$ dimensional Normal Distribution having location parameter $\mu_{10\times 1} = (1,1,\dots,...
1
vote
2answers
272 views

Differentiation of an unknown function

I have to take the partial differentiation of an unknown function. For example, take the unknown function to be $g(x)$. Then it's derivative w.r.t $x$ is $g'(x)$. By default, Mathematica ...
2
votes
0answers
251 views

Getting error messages from a Plot expression containing an integral [closed]

This should be a really simple question, but I just got confused. Plot[Integrate[x, x], {x, -5, 5}] Integrate::ilim: Invalid integration variable or limit(s) ...
-2
votes
1answer
79 views

Help me with sum [closed]

good morning $$ \begin{array}{l} \left\{ \begin{array}{l} u_0 = 1 \\ u_n = 3 - \frac{4}{{1 - v_n }} \\ \end{array} \right.\quad \\ \;v_n = \left( { - 3} \right)^{n + 1} \\ s = v_{1436} ...
6
votes
2answers
663 views

Integral does not converge (when it should) [duplicate]

Bug introduced in 10.0 and fixed in 10.0.2 I was looking at integrals like: Integrate[HermiteH[50, x]*Exp[-x^2], {x, 0, Infinity}] which gave me a "does not ...
0
votes
1answer
108 views

Operator product

I have this problem: I define the gradient operator as: gradiente = {D[#, x] &, D[#, y] &, D[#, z] &} Now, if I want to apply the operator on a ...
0
votes
1answer
76 views
0
votes
1answer
152 views

Integrating spherical harmonics

I am trying to simply check the integration/normalization condition on the SperhicalHarmonic functions that are built into Mathematica. So basically, I just want to check that the following integral <...
2
votes
1answer
124 views

Double integral over normal pdf gives inconsistent answer

I am using Integrate function to do the following double integral ...
1
vote
1answer
82 views

Performance boost for definite integration with symbolic limits

In Mathematica, the symbolic definite integration ...
2
votes
0answers
86 views

Evaluating an integral

Here is the input: Integrate[E^(2 I*t)/(2*Pi*(E^(I*t) - z)), {t, 0, 2 Pi}] and here the output: ...
2
votes
0answers
117 views

How can I do multiple (double) integral with periodic boundary conditions?

I am trying to do a simple double integral $\int_0^l \left[\int_0^l h^3 h_{xxx}dx\right]dx$ with periodic boundary conditions $h[0,\tau]=h[l,\tau]$ $h_x[0,\tau]=h_x[l,\tau]$ $h_{xx}[0,\tau]=h_{...
0
votes
2answers
125 views

Derivative after numerical Integration

So I have a function that I want to numerically integrate with respect to temperature, so the integral changes depending on the value of temperature being used as it is one of the bounds. After ...
2
votes
1answer
134 views

Limits and transformation rules

I can calculate this limit: Limit[(c^(1 - g) - 1)/(1 - g), g -> 1] and obtain the correct value: Log[c] but if I try ...
0
votes
2answers
134 views

What is the simplest way to plot x^(1/3) on the interval [-3,3]?

My question is: What is the simplest way to plot x^(1/3) on the interval [-3,3]?
4
votes
2answers
124 views

Adding assumptions changes the result of Integrate

When I try to calculate the following integral: ...
2
votes
1answer
128 views

How to avoid complex terms in the results given by Integrate?

As shown by In1, integrating Sqrt[x^2 - 1]/(a + x^2) (1/x) returns a result with a complex terms; however, after I make a ...
2
votes
0answers
210 views

NDSolve is running an extremely long time: how can I save the existing data?

I am trying to solve a PDE by the following code. It takes 1 hour or so to reach t=25.72 but about 20 hours to reach t=25.72404031638060174049337306126853310997. Actually, the time step is extremely ...
0
votes
1answer
162 views

Replacement of terms/Pattern matching involving products of derivatives of a function

In delving into Ramanujan summation, I'm trying to get a hold of the relations of the form $$\sum_{n=0}^\infty f(n)=\dfrac{h\frac{d}{dx}}{{\mathrm{e}^{h\frac{d}{dx}}}-1}\int_{0}^\infty f(x)\,{\mathrm{...
4
votes
4answers
509 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 ...
6
votes
1answer
249 views

Incorrect evaluation of integral involving a DiracDelta, whose argument has infinitely many zeros

Let's say $X>0$ is a random variable with probability density $p_X(x)={\rm e}^{-x}$. Define the random variable $Y=\sin(X)$. From the transformation theorem for probabilities we know that its ...
1
vote
1answer
108 views

Strange timings of integrals involving Hermite's polynomials

I have used Mathematica to calculate tunneling for quantum harmonic oscillator. The code is simple: ...
1
vote
2answers
107 views

Integrating a compound expression

I have an integral of the form I[r]=∫(arExp[-r]-brSin[k(r-d)]Exp[-r])BesselJ[0,kr]dr where Besse1J[0,kr] is the modified ...
2
votes
1answer
149 views

Mathematica is able to compute an indefinite integral but not the corresponding definite one

I'm trying to compute the following definite integral (μ is a parameter): ...
8
votes
2answers
702 views

Probability: Calculating a multiple integral

Find the value of $P[\Pi_{i=1}^{10}X_i > C]$ for $C=2,5$, where $X_{10\times 1}$ is a random vector with $10$ dimensional Cauchy Distribution having location parameter $\mu_{10\times 1} = (1,1,\...
5
votes
3answers
410 views

Bug in complicated Limit

The code Limit[Log[2 - Sin[x]*Cos[x]], x -> Infinity] outputs Interval[{0, Log[3]}] in Mathematica 10.0.2.0 . It should ...
3
votes
2answers
652 views

Color a Function by its Derivative

Mathematica's ColorFunction seems to struggle with coloring a function "y" by its derivative D[y,x]. This is a seemingly simple task, but Mathematica can't handle it. While I can certainly evaluate ...
4
votes
1answer
112 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 ...