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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Approximating exponential generating function (EGF) from values of generating function (OGF)

I have a function that can evaluate ordinary generating function, and need to construct an approximation to the exponential generating function by using a few calls to the ordinary generating function....
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On a formula in Wolfram alpha

I need to understand a formula for the Hypergeometric function $_3F_2(a,b,c;d,e;1)$ I searched and got the following from Wolfram alpha ( see:- second formula in https://functions.wolfram.com/...
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Analytical Solution in Generalized Heat Equation

Given the following problem $$u_{t}-\frac{v}{2}\cdot u_{xx}+\frac{v}{2}\cdot x^2 \cdot u(x)=0 $$ $$u(x,0)=f(x)$$ Where $ f(x)=y_1(x)+0.2y_4(x)+0.01y_6 (x) $ $y_n$ are the Eigenfunctions that defined ...
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Intersection points of two-variable polynomials

Let $h$ and $g$ designate two multivariable polynomials: $$\!\!h(x,y)=-96\! \left(32 x^2\!+\!8 x (3\!-\!8 y)\!+\!40 y^2-28 y+5\right) \left(10 x^2+x (4 y-2)+\quad\quad\quad\\ \quad\quad+(1-2 y)^2\...
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Why do MMA13.2 and MMA12.3 give different results of the same integral? [closed]

MMA13.2 f13[z_] = Integrate[( L W z (L^2 + W^2 + 8 z^2))/((L^2 + 4 z^2) (W^2 + 4 z^2) Sqrt[ L^2 + W^2 + 4 z^2]), z] // FullSimplify MM12.3 ...
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1 answer
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Assume asymptotic value in a limit?

Is it possible to give Mathematica a limit about's function at infinity? For example I would like to ...
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Fractional Differential Equation

I am facing the following problem. If u[x_,t_]=t+y[t]x, Then, D[u[x,t],t] Gives us 1+xy'[t]. But if I use the command for Fractional Derivative like FractionalD [u[x,t],{t,a}], then Mathematica print ...
2 votes
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Simplify or FullSimplify doesn't work on Gamma function [migrated]

I have this expression: Gamma[n - 0.691 + 2.35 I] Gamma[n - 0.691 - 2.35 I] which returns real values for all $n$. However, neither Simplify nor FullSimplify can ...
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How can we calculate an integral when Integrate does not converge? [closed]

I tried to calculate the probability density of a wavefunction, but got not result. Does anyone have an idea of an alternative approach to get the probability density of it? Here are the details: <...
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Solving analytical integral

I am trying to solve the following integral \begin{equation} \left[\int_{0}^{t/2}d\tau-\int_{t/2}^{t}d\tau\right]\left[\int_{0}^{t/2}d\tau'e^{-\left|\tau-\tau'\right|/\tau_{c}}-\int_{t/2}^{t}d\tau'e^{-...
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4 answers
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Evaluate using Mathematica or otherwise the value of $f'(0)$

If we have$$f(a)=_3F_2\left(\frac{1}{2},\frac{a+1}{2},\frac{a+2}{2}; \frac{a+3}{2},\frac{a+3}{2};1\right)$$ where $a\geq 0$ and $_3F_2$ is the Hypergeometric function with unit argument. Evaluate ...
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Series expansion using binomial theorem in Mathematica

The series expansion of $\displaystyle\left(1-\frac{a}{x}\right)^{1/3}$ using the Binomial theorem is given by $$\displaystyle\left(1-\frac{a}{x}\right)^{1/3}=1-\frac{a}{3x}-\frac{a^{2}}{9x^{2}}-\...
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I can't solve this problem [closed]

$$\sum_{k=1}^{\infty}\ln(\frac{(k + 1) (k + 3)}{(k + 2) ^ 2})$$ The answer is = ln(2/3) Can you please help me?
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1 answer
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Modified Rational approximant of a polynomial (v2)

ok this time I will try to post only the essential information, I hope it doesn't end like the previous thread. the objective is similar to the previous one but now I cannot use the Padèapproximant ...
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Calculate wavelet coefficients with Mathematica

Let $\operatorname{rect}(x)$ be a constant function equal to 1 with support $[-1 / 2,1 / 2]$. Define a step function using the sequence as $$ f(x)=\sum_{k=1}^4 a_k \operatorname{rect}(x-k)$$ where $...
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1 answer
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Green function and differential equation

I'm trying find the Green's function for a special operator. I have the homogeneous equation x (x - 1) y''[x] - 2 I Zm y'[x] - L(L + 1) y[x]==0 for $x>1$ ...
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3 answers
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Evaluate $\int_{0}^{\pi/2} \cos^a (x)\sin(ax) dx$ using Mathematica

Consider the integral $$I(a)=\int_{0}^{\pi/2} \cos^a (x)\sin(ax) dx$$ where $a\geq 0$ I am trying to evaluate the above integral using Mathematica using the following code : ...
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Why does this finite integral blow up in Mathematica?

I am having some trouble understanding the output of a particular integral. I think I am misunderstanding something about how Mathematica is used, or about how the Integrate function works, and I ...
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3 answers
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Evaluate $\int_{0}^{\pi/2} \cos^a (x)\sin(bx) dx$ using Mathematica [closed]

Consider the integral $$I(a,b)=\int_{0}^{\pi/2} \cos^a (x)\sin(bx) dx$$ where $a\geq 0$ and $b\geq 0$ I am requesting a code in Wolfram Mathematica so as to write the non negativity restrictions of $...
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Mathematica doesn't seem to be able to compute the Fourier transform of the Haar orthonormal basis over $L^2(\mathbb{R})$

Consider F[x_, n_, m_] := 2^{n/2}*WaveletPsi[HaarWavelet[], 2^{n}*x - m] A paper I am reading uses a different convention of the Fourier transform than Mathematica'...
1 vote
1 answer
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How to take the derivative of a function in `Im[]`?

I tried to take the derivative of a long expression containing $\text{Re} \text{Li}_3(e^{2\pi ix})$, the trilogarithm while $x$ is real. Since the real part function is linear ,it should be like this $...
3 votes
1 answer
101 views

Is this a bug in IntegrateChangeVariables?

Bug introduced in 13.1, persisting through 13.2.1. ...
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1 answer
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Could Mathematica analytically work with operators (such as $\nabla$) and vectors?

I have the following expression: $$(\nabla+\vec{P})^2$$ where $P=(r\cos\theta sin\phi,r^4\sin\theta,r\cos^3\phi)$ I would like to open the brackets and simplify the expression. Are there in ...
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How to print particular cases with Integrate?

I would like to perform the following simplification: ...
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Calculating double integrals from a very big function

I need to compute integrals (for different value of i and j) of a function named se in this post. We need to double differentiate from se related to i and j and then compute double integrals over the ...
2 votes
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Efficient powers of DPR1 matrices

I'm looking to compute the following quantity for $A$ where $A$ is a convergent positive definite $d\times d$ diagonal + rank1 matrix (DPR1). $$f(s)=\operatorname{Tr}(A^s)$$ Earlier answer answer by ...
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1 answer
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Is there any hope to compute analytically $y=f(x)$ for which $g(x,y)=0$?

I have a two-variable function $g(x,y)$ with $0<x\leq\frac{\pi}2$ and $\frac32<y<5$. I am looking for $y=f(x)$ for which $g(x,y)=0$. Using ContourPlot <...
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Functions for translating a vector from one coordinate system to another

I have a vector $\vec{H}$ with components in Cartesian coordinate system $(x,y,z)$: $\vec{H}=(0,0,H)$. I would like to obtain coordinate of this vector in Spherical coordinate system $(r,\theta,\phi)$....
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Making subscript partial derivative notation work using Notation` [duplicate]

I want to write derivatives like $u_{xx}$ and have this be parsed as D[u,{x,2}]. Immediately the Notation` method came to mind, though I'm not sure how I could implement this. I thought of using ...
1 vote
2 answers
134 views

Doesn't analytically integrate sensibly let alone correctly

(This is induced x velocity on vortex sheet y=1+gtax/g+tx by strength 1+ac/X) ...
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1 answer
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How to use Slater Type Orbitals as a basis functions in matrix method correctly?

This question is a continuation of my previous series of questions about basis functions. I would like to find the minimum energy of Coulomb potential motion using matrix method. $H=-\frac{1}{2}\Delta-...
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1 answer
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Mathematica calculates the parameter value when the first derivative is 0

I want to get the value of the corresponding parameter t when dil=0 or dvc=0. What method can I use? ...
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Asymptotic returns expression unchanged instead of first-order approximation

I'm using Asymptotic to get first-order expansion of expr=Gamma[0, s/10] - Gamma[0, s], and it returns ...
2 votes
2 answers
427 views

Bug in integrate in in 13.1.0?

Bug introduced in 13.1, and fixed in 13.2. I'm trying the following locally (13.1.0 for Mac) and getting incorrect result ...
1 vote
0 answers
56 views

Potential Bug of Floating Point Notation in an Integrated Radical

Mathematica version 13.2, tested on MacOS and Windows. I recently stumbled into what I think is to be a bug with symbolic integration with radicals. Upon testing, I found some cases that I feel ...
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Calculating an integral with a seemingly complicated integrand

I would like to calculate the following integral: $$ I=\int_{0}^{1} \min (1,(u-1)^{n-1}(u-v)^{n-1}) dv $$ where $u$ is a constant taking values between $1$ and $2$ and $v$ is variable whose range is $(...
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Double integral problem with infinite integral

double numerical integral as show in the follow: $-\frac{4 \sqrt{\frac{3}{8}-\epsilon ^2} \text{sgn}(w) \sin \left(\frac{3 w}{2}\right) \cos (100 w) \left(\frac{4 \epsilon \left(-\frac{10000}{3} \...
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Why complementary solution of an ode changes when adding term on the RHS?

I found a case where AsymptoticDSolveValue gives correct solution to a second order ode when the RHS is zero. i.e. complementary solution is correct. As we all ...
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1 answer
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Why two the same integrals give different values?

Why two the same integrals give different values? ...
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1 answer
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How to find the difference equation from system response or primitive equation?

If I have a system response or the primitive equation , how can i find its difference equation by mathematica? eg: ...
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3 answers
102 views

Why earlier terms generated from AsymptoticDSolveValue change when increasing the order?

I was comparing my hand solution with Mathematica's. I noticed the particular solution terms generated for a Frobenius series solution change depending on the order. This should not happen. As when ...
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How to solve double integral involving Bessel functions?

I want to solve the following double integral 'p' which involves Bessel function of first kind and of order zero. By solving I mean, how can I get a simple expression with only k and R.
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How to solve boundary value problem?

Is it possible to solve a boundary value problem like this? ...
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How to compute the residue of $f(z)=\frac{n/z}{z^{n}-1}$ to be $-n$ at $z=0$? [duplicate]

How to get residue of $f(z)=\frac{n/z}{z^{n}-1}$ with correct residue of $-n$ at zero? It works it out just fine at 1 with: Residue[(n/z/(z^n - 1)), {z, 1}]
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FunctionDomain and FunctionRange for x^2/x returns "True"

I'm very concerned. As far as I know, the output should be: $x < 0\ ||\ x> 0$ $y < 0\ ||\ y> 0$
1 vote
3 answers
157 views

Speeding up the time of integration calculation

I am going to integrate from Chebyshev orthogonal functions that is in the form of: ...
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0 answers
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Problem installing Rubi on Mac OS X ARM

I installed the Rubi package by using ResourceFunction["GitHubInstall"]["RuleBasedIntegration","Rubi"] but when I use it, it only ...
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Incorrect result of InverseFourierTransform

When executing in 13.2 on Windows 10, the command FourierTransform[DiracDelta@Cos[x], x, s] results in ...
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2 answers
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Help regarding an integral in about certain probability [closed]

In the following $V$ is an uniform variate in $(0,1)$ ,$T$ is another independent random variable with CDF $t^{n-1} $and $u>1$ is a parameter.A friend sought help in an assignment in the ...
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1 answer
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How to get rid of curly brackets from a function output? [closed]

The following codes of ode provides solutions for x1(t),y1(t),and v(t) as interpolating functions. ...

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