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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19
votes
3answers
2k views

How to deal with complicated Gaussian integrals in Mathematica?

As we know, for most Gaussian integrals, we can get the analytical result. Now I have many Gaussian integrals to treat, which have the following general form, ...
0
votes
0answers
40 views

Problem with indefinite integral

How do I calculate this integral with mathmatica? integrate[(Exp[-x/a]Exp[-p x/a])/(x^2 (1-(a/t)^2 Exp[-2 x/a])^(1/2)),x] a, p, t are constants
0
votes
2answers
168 views

How can I solve this integral analytically?

I want to solve the following integral: Integrate[( E^(-ω/Λ) Sin[((ω + Ω) t)/2] Sin[(ω t)/ 2])/((ω + Ω)/4) , {ω, 0, ∞}] But unfortunately Mathematica cannot ...
-2
votes
1answer
55 views

Finding unit vectors which makes an angle 60 degrees with both {1, -1, 0} and {1, 0, -1} [closed]

Find unit vectors which makes an angle 60 degrees with both vectors {1, -1, 0} and {1, 0, -1}.
40
votes
2answers
734 views

Why does Mathematica report that $\int_1^\infty\frac{\sin(\sqrt{x})}{\sqrt{x}}dx$ = $2\cos(1)$?

Bug introduced in 7.0 or earlier and persisting through 10.4 or later Mathematica 10 gives the following very odd result, ...
6
votes
1answer
86 views

How correctly use WhenEvent to detect when gradient is over a certain value?

So I wish to stop integrating my PDE when the spatial gradient is larger than some value, let's say 10. That is to say, my WhenEvent condition wants to be stop when the the maximum gradient over all x ...
0
votes
1answer
53 views

Evaluate as much of an integral as possible [closed]

Consider the integrand $\qquad \cos(x+\sin^{-1}(x)) = \sqrt{1-x^2} \cos(x) - x \sin(x)$ If I ask Mathematica to integrate $\cos(x+\sin^{-1}(x))$, Mathematica is unable to do so analytically (due to ...
4
votes
3answers
231 views

How can I plot a paraboloid?

How do I plot a function for a paraboloid? Im putting together surfaces to model lipstick. I need the paraboloid for the top part and then I'll be cutting the paraboloid at angle with another surface. ...
3
votes
2answers
106 views

Correct behavior of Integrate involving logarithmic cut, or possible bug?

I just discovered that Integrate with simple poles in its integrand sometimes gives different results depending on number type of its limit (Exact vs Inexact). ...
8
votes
0answers
99 views

Strange kernel segmentation fault in Integrate

Bug introduced in 8.0 or earlier and persisting through 10.4 Define tointegrate = D[V[y1[t], y2[t]], y2[t]] D[V[y1[t], y2[t]], {y1[t], 2}] and try to find its ...
15
votes
1answer
262 views

Integrate returns imaginary answer for smooth, real function

Bug introduced in 7 or earlier and persisting through 10.4 I'm trying to evaluate the integral: $$\int_0^{\infty} \frac{1}{4 b \sqrt{\pi} r} e^{-(b-r)^2}(e^{4 b r} - 1) \mathrm{d}r$$ with ...
5
votes
2answers
204 views

RegionPlot of ImplicitRegion is incorrect

Bug fixed in 10.4. Why does RegionPlot of the ImplicitRegion give an incorrect plot? Also, will how does this affect ...
7
votes
3answers
364 views

Where did I go wrong with my implementation of the trapezoidal rule?

One method for doing quadrature, called the trapezoidal rule, improves accuracy by connecting the points on the curve corresponding to the points of subdivision with line segments, forming trapezoidal ...
2
votes
0answers
88 views

How to Mellin transform a complicated Log integrand?

I got a question concerning an integral. I need to know the analytical expression. I have to Mellin transforn a function and the integral is then sth. like this: $$ \int x^{N-1} \frac{Ln(a -x)}{1-x} ...
3
votes
1answer
55 views

Evaluating function

I'm trying to evaluate this function, which is gradient of integral with variable limit of integration: $$ u(x,y,t)=\nabla(\int_{v(x,y)}^\infty \frac{e^{-b(t-\tau-c)^2}}{\sqrt{\tau^2-{v(x,y)}^2}} ...
1
vote
1answer
103 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[2^i*Binomial[n-i-1,2*n/3-1],{i,0,n/3}] but I ...
23
votes
2answers
528 views

Symbolic derivatives are being calculated numerically

Just found the following while debugging a problem. Mathematica is calculating the derivative of IntegerPart[x] in some odd way: ...
0
votes
1answer
38 views

Limits at infinity not neglecting added constants [closed]

This is such a simple question that I'm sure it must be answered somewhere, but I haven't had any luck finding an answer. I am trying to simplify expressions of the form: $$f(x) = ...
1
vote
1answer
73 views

How to calculate the derivative at a point of a non-integrable multivariable function?

I want to integrate this function with respect to θ1 and then take the derivative of that function with respect to n. Since the ...
3
votes
2answers
168 views

Use Meijer-G function to represent elementary functions

I want to represent these elementary functions: $x^{2}\sqrt{x}$, $\sin{4x}$, and $x\ln{x}$ as cases of MeijerG. What arguments should I give to ...
1
vote
1answer
70 views

The assumption in integration does not work?

I am new to mathematica. Here is the question. I try to do the following integration with the assumption that k1<0, k2<0 and k1,k2 are real number. However, the result includes \sqrt{k1}? ...
1
vote
1answer
154 views

Derivative of a symbolic matrix

I need to write an elasticity equations as: {{a*∂/∂x,b*∂/∂y},{c*∂/∂x,d*∂/∂y},{e*∂/∂y,f*∂/∂x}}.{{u},{v}} where a,b,...,f are ...
0
votes
1answer
52 views

Same integral gives different results with assumption

b = 1 k = 12560 r = 500 res = Integrate[Exp[I*k*x*Sin[o]]/(r + x*Sin[o]), {x, -b/2, b/2}] resab = Abs[res]^2 Plot[resab, {o, -Pi/20, Pi/20}, PlotRange -> Full] ...
0
votes
0answers
77 views

How to evaluate this integral

I am trying to evaluate the following integral in Mathematica: ...
8
votes
2answers
242 views

Can your Mathematica do the following integral?

Backslide introduced in v10 and persisting through v10.3.1. I was trying to do the following integral in Mathematica: ...
2
votes
1answer
78 views

How to plot linear term of a function expanded by Taylor seires in Mathematica?

I am quite new in Mathematica and I am trying to plot a linear term of a particular function in Mathematica. I have tried something like this: ...
1
vote
2answers
72 views

Problems in animating graphs and its extrema

I tested to animate 2D graph and its extrema for simple case using Mathematica. ...
7
votes
2answers
410 views

Need help with the Lagrange multiplier method

Well I am new to Mathematica and got really stuck solving this problem. In class, I was ask to use the Lagrange multiplier method to find the maximum and minimum value of $f(x,y) = x^2+y^2$ which ...
-2
votes
1answer
66 views

How would I integrate cos (x^2) using the trapezoidal rule with n=4? [duplicate]

I need to integrate the function I stated above, with the trapezoidal and midpoint rule. It has the integration bounds of 0 to 1. N is the amount of rectangles I should be using to find the value of ...
0
votes
2answers
89 views

Using Mathematica to find the areas described by polar curves

I am working on a project and I need to find the area of the surface inside the polar-curve $r=2\cos(\theta)$ and outside the polar-curve $r=1$. The graph is pretty straightforward: just two ...
4
votes
2answers
348 views
5
votes
5answers
157 views

How to find the derivative value at $(\pi,0)$ for this implicit function $n$ times?

I am trying to take the implicit derivative at $\sin(x+y)+\sin(x)=y$ and substitute $x=\pi$ and $y=0$ at least 6-7 times since I need to find the Taylor series for this function. Since I barely ...
0
votes
1answer
71 views

Integrate over FindRoot solutions

I have a function of bivariate normal PDF and its marginals defined as ...
1
vote
0answers
40 views

How obtain the n-th derivative of a function with an implicit variable? [duplicate]

I'm interested in obtain the analytical n-th derivative with respect to the variable $x$ of an implicit function like $V(h(x))$ where $h(x)$ is an implicit equation and can not be obtained the ...
1
vote
0answers
98 views

Implicit differentiation dx/dt? [closed]

I am trying to find $\frac{\partial t}{ \partial y}$ as a function of $x$, $y$ and $\frac{\partial t}{\partial x}$ if both $x$ and $y$ depend on $t$ for the function ...
-8
votes
1answer
72 views

How to partially differentiate tu[t]u'[t] with respect to t [closed]

How can I partially differentiate tu[t]u'[t] with respect to t in Mathematica?
-2
votes
1answer
62 views

Wolfram Mathematica ONLINE does not give me the solution for the following integral [closed]

I have some problem in Mathematica and I could not evaluate the following integral. There is a closed-form solution for the following. Can someone please give me the result. ...
2
votes
2answers
149 views

Revolve a non one-to-one function about the y-axis

I am attempting to revolve $f(x)=-x(x-3)(x-2)^2$ about $x=0$. What I have come up with is that I need to redefine each interval between the relative extrema as a separate function, and then treat it ...
0
votes
1answer
44 views

Integrating Poisson distributions [closed]

I'm trying to do the following Integrate[Exp[-λ] λ^k/k!, {k, 0, ∞}] The answer should be 1 but Mathematica is unable to do it. Same story when I try the ...
4
votes
1answer
157 views

Computing the Caputo fractional derivative of a polynomial

In fractional calculus, the Caputo derivative of a monomial has the following form: $$\operatorname{\mathit D}_t^\alpha\,t^\beta = \frac{\Gamma(\beta+1)}{\Gamma(\beta-\alpha+1)}t^{\beta-\alpha}$$ I ...
25
votes
5answers
6k views

Multivariable Taylor expansion does not work as expected

The basic multivariable Taylor expansion formula around a point is as follows: $$ f(\mathbf r + \mathbf a) = f(\mathbf r) + (\mathbf a \cdot \nabla )f(\mathbf r) + \frac{1}{2!}(\mathbf a \cdot ...
1
vote
1answer
42 views

Having trouble taking derivative of a function and assigning it to a new function [closed]

I am trying to take the derivative of a function and assign it to a different function name, but I am getting an error message (I think). Here is my Mathematica input: ...
3
votes
2answers
150 views

How to get the training error and the validation error using Classify function with SVM

I am using Classify[] function with Support vector machine model. I need to analyze the quality of results by analyzing the training error and the validation error ...
0
votes
0answers
51 views

Integration of a Gaussian function in kernel density estimation

I came recently across an odd result of mathematica, which so far I cannot explain. In a standard kernel density estimation (example http://www.ssc.wisc.edu/~bhansen/718/NonParametrics1.pdf) the ...
4
votes
2answers
147 views

Derivative of piecewise function

I tried to find $f'(0)$ of this function: $$f(x) = \begin{cases} x\cdot\sin(\frac{1}{x}) & \text{if $x\ne0$} \\ 0 & \text{if $x=0$} \end{cases}$$ This is what I tried in Mathematica: ...
0
votes
1answer
37 views

Efficient calculation of leading order behavior of rational function under a limit?

Considering a rational function, as for example myExpr=(x+3)(2+5x+6x^2)/(x(x-5)(x+20)) I would like to have a function that tells me the leading order behavior ...
5
votes
1answer
74 views

Behavior of D and Derivative when acting on functions with Rules as arguments

I was (pleasantly) surprised that when a Derivative is taken of a symbolic function for which the last few arguments are Rules, ...
5
votes
1answer
101 views

Somewhat Irreproducible Integrate Results

Backslide introduced in v10 and persisting through v10.3.1. In the course of considering question 102922, I encountered erratic results from a particular integration. It is illustrated as follows. ...
3
votes
1answer
109 views

Mathematica Can Find the Primitive Function But It Cannot Find the Closed Form for Corresponding Definite Integral

Consider the following function f = 1/2 (-3 + 4 u) Sqrt[-u (1 + u) (-1 - u + u^2)] where it is assumed that ...