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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2
votes
1answer
28 views

How to mark non-pre-defined functions as functions of a certain variable during calculation of derivative?

If I have some calculation $X=v\cdot\delta$ and I want to calculate $\frac{\partial}{\partial v}X$, the result according to Mathematica is $\delta$. Now, I want to define $\delta$ as a function of $v$ ...
0
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0answers
32 views

Why do I get the error message as integral does not converge?

Can anyone please help me to find the integral value of the following expression, Integrate[(x - t)^(-1), {t, 0, x}] It will be of a big help to me.
3
votes
1answer
73 views

Evaluating an integral combining a Bessel function with some other functions [closed]

How can I evaluate the integral of $j_1 ^2(x)\exp(-bx)/x$ from 0 to ∞?
5
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1answer
73 views

Variable dimensional integrals

I need to Integrate a function, but let the number of dimensions vary. Integrate[Product[Exp[-( t[[i]] )^2], {i,1,4}], options] ...
2
votes
2answers
104 views

Evaluating FourierTransform like integral manually

I am trying to evaluate Integrate[x^2*Exp[I k (x - 1)], {k, -∞, ∞}, {x, -∞, ∞}] Since $\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{ik(x-1)} d{k}$ is $\delta(x-1)$, ...
1
vote
1answer
36 views

Limit of an integral and hypergeometric function

I want to evaluate the following integral: Integrate[Sin[θ]^(D1 - Nc - 1)/(A Cos[θ] - I ϵ)^(N1 - Nc), {θ, 0, π}, Assumptions -> A > 0 && ϵ > 0] ...
2
votes
2answers
141 views

Compute unstable integral with high precision

I want to calculate $$ \int_{\mathbb{R}^n} e^{-t_1^4-t_2^4- ...-t_n^4-(1-t_1-t_2-...-t_n)^4} \text{d}t_1 \text{d}t_2 ... \text{d}t_n $$ with the highest possible n. I tried with ...
3
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4answers
443 views

How can I simplify $\lim_{n\to \infty } \, \int_0^n e^{-x} \left(1-\frac{x}{n}\right)^n \, dx$?

How can I simplify $\lim_{n\to \infty } \, \int_0^n e^{-x} \left(1-\frac{x}{n}\right)^n \, dx$ ? Limit[Integrate[(1 - x/n)^n/E^x, {x, 0, n}], n -> Infinity] ...
5
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3answers
494 views

How do I find and plot the intersection of these three surfaces?

I know how to find the region of intersection of these cylinders using integration, but how would I plot that in mathematica? This is what I have ...
1
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1answer
82 views

Problem in Integration

I want to calculate both of these integrals: Integrate[1/Sqrt[c x + Sqrt[ b x + x^2] + a], x] and ...
2
votes
1answer
93 views

Calculation of sum $\begin{aligned}\sum_{k = 1}^{n - 1}\end{aligned}\left(1+\cos\left(\frac{k\,\pi}{n}\right)\right)^n$

Having established that Mathematica cannot calculate the following summation: sum = Sum[(1 + Cos[k Pi/n])^n, {k, 1, n - 1}] I implemented the classic "plan B", ...
2
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1answer
88 views

When to trust Integrate?

I'm trying to get an analytical expression for the following integral: ...
3
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2answers
167 views

On Certain Characteristic Function

I am trying to find the characteristic function for Johnson's SU distribution by integrating the probability density function with Exp[I*t*x] but Mathematica is ...
0
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1answer
62 views

Is there any way to do this integral without expanding the exponential power?

Integrate[ x^(1/2) Exp[-10.7* c/x] Sqrt[(1 - (125/1000)^2) (1 - (2*4.18/125)^2) - (1 - (2 x/ 1000))^2], {x, 5.03, 994.97}] or can i do this indefinite ...
4
votes
3answers
108 views

Problem with minimization of $x^z$

I have plugged in the following problem Minimize[{x^z, z >= 1, z <= 10, x >= 1}, {x, z}] Can anyone explain please why the software does not provide the ...
0
votes
1answer
38 views

How to deal with the derivative of an variable upper limit integration by the symbolic form?

I need a rusult with the following form D[Integrate[f[t], {t, low[x], upper[x]}], x] (*Out[]=-f[low[x]] (low^\[Prime])[x]+f[upper[x]] (upper^\[Prime])[x]*) My ...
0
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1answer
36 views

Interesting output in Integrate [closed]

I found interesting behaviour, best illustrated in the following example. When trying to evaluate Integrate[1/(d^2 - 1), d] Mathematica 11.3 gives the following ...
1
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0answers
48 views

Fitting a Curie-Weiss law function with a diverging point to a data

I would like to fit a function to some experimental data such that the extracted value of Curie temperature (Tc) describes the temperature at which the response flattens out to a constant value. ...
0
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0answers
23 views

Problem with integrating special function under assumption

i found different results when integrating a special function (see below), depending where i place my assumption (x > 0, x0 > 0). The problem is that the two solutions are not compatible. In fact, if ...
0
votes
1answer
64 views

Integrate a function of NDSolve

I'm having this problem of integrating a function that's the solution of a NDSolve multiplied by another function. I basically compute a cycle wich add at each step the quantity ...
-1
votes
1answer
58 views

Coupled PDEs: Wave and String Equations

I need to solve a system of mixed string and wave equations. Omitting some constants it looks like this: $$u_ {\text {yy}} (y, t) - u_ {\text {tt}} (y, t) = \varphi _{t}(x, y, t)$$ $$\nabla _{\{x,y\}}...
1
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0answers
13 views

Eingenvalues and Listplot take a long time optimize

I try to calculate the following eigenvalues of a matrix of N=5 10^3 X5 10^3 and representing it using Listplot as follow ...
1
vote
3answers
80 views

Summation and product term expressed in Mathematica [duplicate]

I have the following term which I would like to express correctly in Mathematica: $$\sum_{i=1}^{m}\frac{1}{\prod_{j=1,j\neq i}^m(\rho_i-\rho_j)}$$ This means that for $m=3$ for example you should ...
0
votes
1answer
63 views

Evaluate integral over quadratic form [duplicate]

What would be the most kosher way to evaluate an integral like this in Mathematica: $$\int_\mathbb{R^3} \exp\left((x-b)^tA(x-b)\right)dx \ ?$$ Here $A$ is a real symmetric matrix and $b$ some vector ...
1
vote
1answer
63 views

NIntegrate a singular function

I am trying to do this integral numerically: ...
0
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0answers
22 views

Assign a Derivate to a new fuction variable

I'm having a new problem. I'm trying to create a new function from a Derivate of another function, but i'm can't getting it. Someone could help me? I'm thinking that is something in the syntax of the ...
-1
votes
1answer
71 views

How to find the maximum of a function with several variables symbolically?

I want to maximize a function with respect to tau1, tau2, and alpha, and it is like the following: V + β1 τ1 - α τ1^2 - β2 τ1^2 + \τ2 - 2 τ2^2 + α τ2^2 I know I ...
2
votes
1answer
82 views

Why is Mathematica returning the original function from FindMaximum? [closed]

I am trying to find the maximum of the following expression: I can not figure out why Mathematica returns me the original expression. I am new to Mathematica and tired to search around but could not ...
1
vote
1answer
53 views

Area of a parametric plot with interpolating functions

I am looking for the area of a parametric plot defined by two solutions (defined as: VVLt & PVLt) of Numerically solved differential equations. The problem is when i'm trying to integrate the two, ...
1
vote
1answer
90 views

How to integrate exponent of sine?

I am trying to integrate Abs|E^ $\sin \theta$| (the absolute value of Euler's constant to the power of sin(theta), and tried the code Integrate[Abs[E^Sin[x]], x] ...
2
votes
4answers
432 views

Integrating an absolute function using Mathematica

I was unsure as to how to integrate an absolute function, such as $\int_0^3 \left|\sqrt{9-x^2}\right|dx$, so I tried: Integrate[Abs[(9-x^2)^(1/2)],x] But it ...
2
votes
1answer
115 views

Mathematica just takes infinite time to solve this

Can You help? I don't even care about the exact solution. I will be satisfied by the series expansion of the result around a=0. I tried to solve by series expansion and the same result .. infinite ...
0
votes
1answer
53 views

Delayed differential equation with convolution

So I have a differential equation : $\left\{\begin{split} & c'(t) = \alpha q(t) c(t)^{2/3} - \beta(1 - q(t))c(t)\\ &\chi'(t) = \kappa c(t) q(t) \\ & q(t) = \int_{-\infty}^t \Big( ...
0
votes
0answers
94 views

Simplify the equation in terms of function and coefficients?

I have two fourth order differential equations and eight boundary conditions. I want to get my result in terms of function (Y) and coefficients (let say P1, P2 and so on). I used Dsolve to get the ...
2
votes
1answer
82 views

Unstable symbolic expectation of gaussian distribution on absolute values?

Context I use mathematica 11.3 and 12. 11.3.0 for Mac OS X x86 (64-bit) (March 7, 2018) 12.0.0 for Mac OS X x86 (64-bit) (April 7, 2019) I am interested in evaluating the following integral ...
0
votes
1answer
98 views

Chain rule by Mathematica

Please, how to evaluate dh(g(x))/dg(f(x)) by the following definitions: ...
2
votes
1answer
75 views

Singular Integration

I am trying to simplify the following integral but getting no answer. Any help in how to get the resulting function as a function of t would be much appreciated? ...
2
votes
0answers
61 views

Getting Mathematica to supply derivatives in standard format [duplicate]

I will like to use the results of Mathematica outputs in formatted text for students. However, I do not know how to get rid of the variable arguments and the use of raised indices to indicate ...
1
vote
1answer
78 views

An optimization problem over two variables

I have the function fun[x_,y_, t_] = Cos[x] + Exp[I y] Sin[x] Sin[t]; such that $0\le {\rm x}\le \pi$, $0\le {\rm y}\le 2\pi$, and $0\le {\rm t} \le 10$. I compute ...
0
votes
5answers
54 views

Expressing D[1/D1, x] in terms of D1 when D1 = 1 + x

For simplicity, taking D1 to be equal to 1 + x, I want to evaluate the derivative of 1/D1 (...
0
votes
1answer
43 views

Integrating in polar differs from simple integration [closed]

I have same line in orthogonal and polar coordinates, it is Cos[t] in first quadrant. But when I integrate those, result differs and I dont understand why: ...
2
votes
0answers
59 views

Issues with symbolic integration

I am trying to do the following indefinite integral, ...
0
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0answers
63 views

How to Solve This Integral?

Mathematica insists that the below integral does not converge, though it has been solved by my peers, so I am inclined to think that it does. q is an arbitrary variable. ...
4
votes
1answer
189 views

How to approximate the partial sum formula of a summation to all real numbers using Mathematica?

I am sure one has to use indefinite sums and the Euler-Maclaurin formula. $$\sum_{x=0}^{n} f(x)=\sum_{x}f(n+1)-\sum_{x}f(0)=\int_{0}^{n+1}f(t) \ dt -\frac{1}{2}f(n+1)+\left(\sum_{k=1}^{\infty}\frac{...
0
votes
1answer
32 views

Limit function and DirectedInfinity as output: Interpretation

I tried to compute the following limit: In[12]:= Limit[-A/2 x - 2 x, x -> +Infinity] Out[12]= DirectedInfinity[-2 - A/2] Does the output tell me that the ...
0
votes
1answer
84 views

DiracDelta is seldom chosen as the answer [closed]

It is well known that Limit[ ϵ/π/(x^2 + ϵ^2), ϵ -> 0] is the Dirac Delta function DiracDelta[x] Yet Mathematica gives ...
1
vote
1answer
35 views

Using rule with derivative

Assuming I have an expression like: D[a[t, r], r] + D[D[a[t, r], r],r] + D[D[a[t, r], t],r] I would like to change variables, by expression everything in terms of ...
1
vote
1answer
65 views

Very different results of NIntegrate using different methods: which one to believe

Consider the following function: ...
7
votes
2answers
764 views

Integral with DiracDelta. Can Mathematica be made to solve this?

This was an exam question. $$ \int_0^{2 \pi} \delta(\sin^2(\theta) -x ) \,d\theta $$ Direct use of Integrate on it does not give the solution. Is there a ...
2
votes
3answers
177 views

Mathematica won't evaluate this integral

The function (in raw input form) and bounds I used are as follows: ...