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
157 views

How does Mathematica calculate integrals? [duplicate]

One can use mathematica to get $$ \int_{-\infty}^{\infty}\frac{\sin^2x}{x^2+1}\ dx=\frac{\pi \sinh 1}{e}. $$ Here are my questions: [Edited:] Is there a way that I can see the intermediate steps ...
5
votes
1answer
383 views

How to solve this Integral equation

D[x[t] - x[t - 1]/(2 E), {t, 3}] + Integrate[E^(-δ)*x[t - δ]/5^t, {δ, 2, 2.5}] == 0 I found solve this problem is hard with Mathematica. I also find a article ...
5
votes
2answers
173 views

Definite integral closed-form expression

Is there a way to get Mathematica yield a closed-form expression (in terms of special functions) for the integral: $$ \int_{0}^{\infty} e^{-a t}\log(t)\log(1+t)\,dt, $$ where $a>0$? The obvious ...
1
vote
2answers
118 views

How to find the limit of a function of two variables

I was wondering if there is a method to find the limit of a function of two variables at a point. I tried the following: ...
-2
votes
1answer
88 views

How would I find the area of a function over a specific interval? [closed]

How would I find the area of a function over a specific interval? For example: $g(x) = e^{5x}$ Find the area of the function over the interval $[-4, 2]$.
0
votes
1answer
136 views

How I can get faster code for computing a fractional integral?

I use of this code Fractional Integral and wrote the following code: ...
1
vote
0answers
58 views

Limit not evaluated [closed]

I hope this question ha not been asked since the limit is evaluated in 1 not at 0 or infinity. I do not know how to ask Mathematica to find the limit as n -> 1 of the following function: ...
0
votes
1answer
74 views

Rotate from 0<=X<infinity [closed]

Find the volume of the solid obtained by rotating the region below the graph of y = e^(−x) about the x-axis for 0 ≤ x < ∞ With that given expression I tried to do it by doing the following code: ...
6
votes
2answers
171 views

Probability of distance of two random points in the unit circle

Inspired by the problem solve this probability problem symbolically which I have already generalized there I'd like to ask here the same question for a unit circle. More precisely: given two randomly ...
0
votes
0answers
49 views

Sum of Infinite convergent series

I have a series as Sum[(x^1 - (x - 1)^1) (b^Log[(1 + (x))]), {x, 1, Infinity}] which is convergent when b=0.3 and ...
0
votes
0answers
21 views

Syntax Error with Expectation over Distribution

I am getting some errors when I try to execute the following code. Here, the expectation is taken over $g$ which has Nakagami Distribution. ...
-1
votes
1answer
87 views

Double integrals in quantum chemistry [closed]

In quantum chemistry we often encounter integrals like this: I can't understand what is the problem. Can anybody help me?
0
votes
0answers
10 views

Inverse Laplace Transform of an Infinite Sum [migrated]

How to find the Inverse Laplace Transform of the following expression $$1+\frac{-Xs^{2/a}-Ys^{3/b}}{1!}+\frac{(-Xs^{2/a}-Ys^{3/b})^2}{2!}+\cdots$$ Any approximation is also okay... Here $a$ and $b$ ...
0
votes
1answer
62 views

Why does Mathematica 10.3.1 fail me on : Integrate[(x^3) [Log[2 Sin[x]]]^8, {x, 0, Pi}] [closed]

All it gives me is a pretty restatement of my original request. When I use WolframAlpha PRO, I get a result of 624510 and a visual representation of the area in question. I know the ends of the ...
13
votes
5answers
509 views

solve this probability problem symbolically

Consider a unit square, Pick two points P and Q uniformly at random inside the square, What is the probability that |PQ|>1? I tried solve this problem ...
3
votes
1answer
106 views

Calling Integrate with many limits of integration

I accidentally discovered that you can put many numbers into the limits of integration of Integrate and Mathematica will still evaluate it without returning an ...
4
votes
3answers
179 views

How to solve this integration? [closed]

I need to find the integration of the following expression ...
1
vote
2answers
90 views
4
votes
1answer
86 views

Am I getting a correct answer with the Dt command?

We are asked to find $dy/dx$ when $x^y=y^x$. Our hand calculations use logarithmic differentiation. $$\begin{align*} x^y&=y^x\\ \ln x^y&=\ln y^x\\ y\ln x&=x\ln y\\ \end{align*}$$ Then we ...
0
votes
0answers
6 views

Proving infimum [migrated]

I have to prove: $x=\frac{3n-1}{n}|x\in R,n \in N$ inf=3 First I have to prove by induction the sequence is growing, but then i got this and I'm stuck: $\frac{3n+2}{n+1}>\frac{3n-1}{n}$ not sure ...
9
votes
3answers
443 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
votes
0answers
5 views

Future Value Calculation with extra payments [migrated]

I am attempting to create a 'savings goal' calculator but am stuck with how (if possible) I can achieve the following:- Calculate the monthly payment amount required to reach a goal given an ...
2
votes
3answers
89 views

How to find nth log derivative using mathematica?

Now I have a function, say $f(k,z)=e^{-kz}(1+kz)$ I want to find the $n$th $\log$ derivative with respect to z. like $(z\partial_z)^{(n)}f(k,z)$ (or $(\partial_{\ln z})^{(n)}f(k,z)$ if you like), ...
8
votes
2answers
806 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 ...
7
votes
4answers
470 views

How to find a tangent line with 2 points of tangency for a curve?

Say I have a function like this: f[x_] := 4 x^4 - 9 x^3 - x^2 + 10; Plot[f[x], {x, -1, 2}] It's obvious that there's a tangent line with 2 points of tangency ...
0
votes
0answers
24 views

Jackson integral applied to a recursion

Two days ago I asked a question about the way to compute a Jackson integral $\int_0^t h(x[s]) d_qs$. But now, I must applied it in the follwing recursion : $ \left\{\begin{array}{l} x_0[t,q] = ...
0
votes
1answer
43 views

Dependency of parameter to find maximum vanishes after simplification

EDIT: It turns out I just made a dumb mistake: Simplify[Exp[x^2]*2^n*Pi*D[maxdist[n,x]/n*(1+Erf[x/Sqrt[2]])^(2-n),x]] A portion of my "factor" is actually inside ...
3
votes
1answer
88 views

Writing a function that finds the difference between two maxima

Suppose i have two functions: $\qquad -2x^2 +5x+3$ and $\qquad -5x^2 +2x+3$ To solve the point where the functions are at maximum i use the command NMaximize ...
3
votes
1answer
109 views

Jackson Integral

Following an old post, I have tried to make a function that calculates the Jackson integral of $q$-calculus: ...
3
votes
2answers
147 views

Computing a binomial series

How can I compute the following series: Module[{n = 10^6}, Sqrt[n] * (4^-n) * NSum[Binomial[2*n - 1, n - k]/((2 k - 1)^2 + Pi^2), {k, 1, n}]] including ...
1
vote
1answer
54 views

Total and partial derivatives [closed]

The following piece of Mathematica code calculates the total derivative of a function $f(x,y(x))$ and compares it with what it should be. However, the result is not zero: ...
0
votes
0answers
55 views

Integrate an expression and simplify the result

I have to find the indefinite integral of the following: $$\int\sqrt{ax^4 + bx^3 + cx^2 + dx + e} dx$$ where a, b, c, d, e are constant of the order 0.0001 or less but always greater than 0. I ...
0
votes
0answers
62 views

Switching from Integrate to NIntegrate

I'm trying to decrease the amount of time that it takes to run the following code: ...
1
vote
3answers
56 views
10
votes
3answers
459 views

Maximize violating constraints

I have Maximize[{(h*10)/(300*(100 - (l^.5 + d^.4 + H^.6))), (l + d + H + h) == 669, l > 0, d > 0, H > 0, h > 0}, {h, l, d, H}] I believe ...
0
votes
0answers
76 views

Setting up and solving an equation

I would like to solve the equations: $$ F(\xi;x,t)=\int_0^\xi u_{0}(\xi)\:d\xi+\frac{(x-\xi)^{2}}{2t} $$ to use in evaluating: $$ u(x,t)=\dfrac{\int_{-\infty}^\infty \dfrac{x-\xi}{t} ...
3
votes
1answer
52 views

Integrating a gradient and matching terms automatically

I am trying to integrate a function $\nabla w(x,y)$ indefinitely knowing both components of $\nabla w(x,y)$. I have a function dw[x_,y_] that I have defined. I then ...
1
vote
2answers
135 views

How to do this integral? [closed]

Is there any workaround to do this integral? Integrate[(x^2+2 x+1+(3 x+1) Sqrt[x+Log[x]])/(x Sqrt[x+Log[x]] (x+Sqrt[x+Log[x]])),x]
0
votes
0answers
33 views

How can I get left-sided-sequence with InverseZTransform

InverseZTransform[z^2/((4 - z)*(z - 1/4)), z, n, Assumptions -> {1/4 < Abs[z] < 4}] (* -(1/15) 4^-n (-1 + 16^(1 + n)) *) the result is not true.It only ...
1
vote
1answer
118 views

Summing infinite series that converge only for some parameter values

The input Sum[d^t,{t,0,Infinity}] produces output 1/(1-d) which is correct for $|d|<1$. But for $|d|\geq1$ the sum does ...
1
vote
1answer
31 views

Mathematica Integral[] return [closed]

I am new to mathematica and while trying to find the double integral of 8-xy over -2 ...
0
votes
2answers
86 views

Expression of an integral

I am doing the following integral in Mathematica with $a>0$: $$\iint e^{-\frac{(x_{1}+x_{2}-2b)^2}{4a}}dx_{1}dx_{2}$$ My code is ...
0
votes
0answers
102 views

Implicit hidden assumptions of Mathematica with Limit

I've stumbled upon the following symbolic limit (a tricky one): Limit[(Sqrt[2] v)/Sqrt[v^2 - c^2 (1 - 1/c Sqrt[c^2 - v^2])], v -> 0] Mathematica gives 0 as ...
20
votes
5answers
5k views

How can I differentiate numerically?

Mathematica has two ways to integrate: Integrate and NIntegrate. But what about D? ...
8
votes
0answers
84 views

Slot number cannot be filled

I got a strange message with the derivative of a two-argument function defined by a definite integral. For example if "f" is defined as: ...
2
votes
1answer
81 views

Getting wrong result when Integrating under an assumption [duplicate]

The simple integral $$\int_0^b \cos\left(\frac{2\pi m(y-\eta)}{b}\right) \cos\left(\frac{2\pi \eta}{b}\right)\mathrm{d}\eta$$ can be easily evaluated by Mathematica as, ...
0
votes
0answers
51 views

Using DeleteCases to ignore term of Product

I'd like to calculate $\int_{-\infty}^{\infty}\mathrm{d}x/(1+x^6)$ through a variation of the residue formula, which is $\int_{-\infty}^{\infty}f(x)\mathrm{d}x=2\pi i\sum \text{Res }f$ for Residues in ...
3
votes
1answer
91 views

Indefinite integral of nested trig functions [closed]

I tried vainly to integrate nested trig functions like Integrate[Sin[Cos[x]], x] Mathematica does not find a solution. At first I thought there was no solution, ...
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, ...