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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20 views

Product and limit of many (>2) matrices

So I have a matrix like this $$ \begin{bmatrix} 1+iA & iAe^{-2ian/N} \\ -iAe^{2ian/N} & 1-iA \end{bmatrix} $$ in mathematica form is like this ...
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24 views

Trapezoidal Rule Apprx. step size n=10 Issue [closed]

I am attempting to use the trapezoidal rule to find an integral approximation. Can someone see where I am going wrong, please? ...
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0answers
40 views

Mathematica with multi-threaded processors [closed]

Recently I became the owner of a PC based on AMD Ryzen 7 2700 Pro processor. This is a multi-threaded processor (8 cores / 16 threads), other specifications are shown below. In addition, 16 GB of RAM ...
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2answers
87 views

Fitting data to equation with one fitting parameter

If I have the following data (x value is composition and y value is temperature): ...
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1answer
53 views

Extract real part of a complex function

I have a problem for extracting the real part of a complex number. The problem is the following: Suppose $f(z)=\frac{1}{z+\frac{\Delta_{1}}{z+\frac{\Delta_{2}}{z+\frac{\Delta_{3}}{z+...}}}}$, in which ...
0
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1answer
138 views

Integrate real function returns complex function [closed]

I want to compute the integral $$ \int_0^c \exp(-cx+x^2) \mathrm{d}x, $$ where $c>0$ is an unknown constant. In Mathematica Version 12.2.0 ...
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1answer
89 views

Evaluating this generalised integral

I have the following integral $$\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\exp \left(a u^2+b v^2+c u v\right) \; dvdu,$$ which returns the following solution: $$\frac{2 \pi }{\sqrt{4 a b-c^2}...
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34 views

Derivative of NDSolve output

I am trying to study the stability of a 2-dimensional discrete system (X, Y) by finding the Jacobian at the systems non-trivial equilibrium (X*, Y*). The functions that map the system from one iterate,...
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0answers
36 views

How to combine long outputs from differentiation in Mathematica? [closed]

I have done some calculations that yield long outputs. I would like to name these outputs so they can be called in a later operation. for instance, output 1, is called W, and output 2 is called R, and ...
0
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1answer
45 views

How can I change the dependent variables of a function?

What I want is to define a function which depends of some variables and then get the expression for that same function when it depends of different variables. My specific case is: (where ...
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0answers
32 views

How to change the variables of a function?

How can I transform: u[x,y,z] to u[x0,y0,zR] Where: z = zR + f; u [x, y, z] = 1/(I [Lambda] z)Integrate[u0 [x0, y0] Exp[(I k)/(2 z ) ((x - x0)^2 + (y - y0)^2)], y0, {x0, -Infinity, +Infinity}] ; u0 [x,...
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1answer
62 views

What results of FunctionMeromorphic are true?

Studying a new command of 12.2 FunctionMeromorphic, I try FunctionMeromorphic[(Re[z] - I*Im[z])/Abs[z]^2, z] ...
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1answer
62 views

Accurate Numerical values of a given functional

Consider the following functional : $$ I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy, s) − F(x −\mathrm iy, s)}{\mathrm e^{2πy}-1}, $$ where $ F(z, s) = \dfrac{\sin^2[π\Gamma(z)/(2z)]}{z^s} $....
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0answers
68 views

Non-convergent integral converging after variable transformation

When I try to integrate the following function with Integrate[...] $$\int_0^1 dx \frac{1}{1-x}\left(\frac{1-x}{x} \right)^i$$ with $i$ the imaginary unit, mathematica says that it is not convergent. ...
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1answer
68 views

How to use Mathematica to evaluate the limit of Exponential Functions? [closed]

How I can calculate a limit using l'Hopital's Rule, limit of (e^-2x +3x)^1/x , as x approaches infinity. How can I do it in Mathematica?
4
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2answers
113 views

`Apply` function doesn't work as expected [closed]

I'm using Mathematica 12.2. I've just got surprised when the following simple line of code does not work properly. This is my input (as simple as it could): ...
1
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1answer
23 views

Substitute a function

In the following code, I'm trying to substitute Sin[x] for f[x]. What's the right way to do it so that f'[x] gets replaced with <...
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0answers
22 views

Replacement of functions bound by sums and integrals in NCAlgebra

I want to use the NCAlgebra package to do some simplification on non commutative expressions involving integrals. For example, one such expression would be $$ I=\left(\int f(x)g(x) dx\right) * h $$ ...
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0answers
32 views

How to write these kinds of integrals in a more compact way?

I wanted to calculate the average distance between two points both lying inside $B^2(\vec 0, R)$, and ended up writing ...
2
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0answers
68 views

Numerical Mixed Partial Derivatives: How to Choose Scale?

For illustration purposes, consider the function $$ f(x, y) = (x - 1)^2 + (y - 1)^2 + \sin[25(x+y)] . $$ The mixed partial derivatives at the origin are $f^{(x,x)}(0,0) = f^{(y,y)}(0,0) = 2$ and $f^{(...
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1answer
92 views

Derivative of -I (Log[-x] - Log[x]) - why is it zero? [closed]

Mathematica gives the derivative of the function -I (Log[-x] - Log[x]) as $0$, but on the real domain the expected result is $\pi\delta(x)$ and on complex domain it ...
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0answers
21 views

Error using ParallelMap to evaluate NIntegrate of numerical derivative ND

I want to evaluate a numerical integration in which the integrand is constructed out of a function f and its derivative f'. <...
2
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2answers
88 views

How to solve this differential equation with a square root?

I've to solve this differential equation but I don't know how to do it. Someone can help me? $\frac{dy}{dx} = \frac{1}{x} \Bigg( -1 - \frac{2a}{\sqrt{a^2 - 3x L f^2 e^{3y}}} \Bigg)$ where a, L and f ...
2
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1answer
45 views

Evaluation of divergent hypergeometric function

I try to calculate the following hypergeometric function: HypergeometricPFQ[{n+1, n+1, n+3/2}, {2n+3}, -1] for several positive natural values $n=0,1,2,\ldots$, ...
7
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1answer
246 views

Incorrect integration from article

Recently I came across an interesting article by Daniel Lichtblau, where the integral Integrate[Abs[x - y]^n, {x, 0, 1}, {y, 0, 1}] ...
1
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1answer
53 views

Plot with three functions and three parameters [closed]

I think my code doesn't have any problem but the program shows me this error and I'm confused! Plot::argr: Plot called with 1 argument; 2 arguments are expected. ...
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0answers
44 views

Parameter sensitivity when integrating in Mathematica [duplicate]

When I try to integrate this MMA freezes: ...
1
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1answer
35 views

Variable substitution when integrating with Mathematica

When I try to integrate this MMA freezes: II = (2 fx R Cos[ϕ] Sin[θ])/(R^2 + ρz^2 - 2 R ρz Cos[θ])^(3/2) ...
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1answer
48 views

How do I plot the derivative of the Cantor Staircase?

I know that the derivative of the Cantor Staircase should be close to 0 since the Cantor Set has a measure of zero. I wanted to graph the derivative of Cantor's Staircase in Mathematica to see if it ...
2
votes
2answers
177 views

How do you get substitution to recognize a derivative?

I have a simple expression involving a derivative: x[t_] = Q/T[t]; Simplify[Derivative[1][x][t] /. Q -> x T[t]] That produces:$$-\frac{x T'[t]}{T[t]}$$I have ...
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0answers
37 views

Wolfram alpha query works online but not through mathematica?

Bit of a bizarre problem I'm having (well, more a question than a problem). I am trying to solve an integral, which I queried using the online wolfram alpha website. The input and output are ...
4
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2answers
143 views

Evaluating an improper integral

So I have this function: $$ \int_{-\infty}^{\infty} \mathrm {sech}(x+s)^{2} {\rm sech}(x)^{2}dx$$ And when I try to integrate it, I can obtain the indefinite integral: ...
7
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3answers
577 views

How can I draw a vector field on a curve?

For didactic purposes (a line integral of a vector field) I'd like to plot a vector field along a curve in 2D and 3D, like in this picture: Mathematica is able to vizualize vector fields. Here is my ...
0
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0answers
70 views

Incorrect outputs of Limit command

Version 12.0 produces a generic result Limit[Cos[a*x], x -> Infinity, Assumptions -> a \[Element] Reals] Indeterminate ...
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0answers
28 views

Numerical integration involving divergence

I was just trying to perform the following numerical integration NIntegrate[1/(x - 5)^2, {x, 0, 10}] The problem I am facing is the divergence at ...
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1answer
55 views

Mathematica does not simplify divergent integrals

I get this as output: $\frac{1}{6} \int_0^{\infty } 6 t \, dt$ Why it cannot cancel the coefficients?
0
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2answers
81 views

How to solve this inequality? [closed]

solve for the smallest N value for this inequality, I tried several online calculators and they are not working.
2
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1answer
60 views

How to compute the limit of a subdivided rotation applied to itself n times?

I have the following function: f[n_, angle_, axis_] := Simplify[RotationMatrix[angle/n, axis]^n]; where n is a positive integer, angle is real and axis is a 3d ...
1
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3answers
138 views

Error with NIntegrate of a two variables integration

This is mathematica code for my integration, ...
2
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1answer
250 views

How do you turn this statement into a Mathematica formula?

I'm reading a book on Nucleosynthesis. It has this formula for the number density of a species:$$n_i=g_i\space e^{\mu_i/T}\int\frac{d^3p}{(2\pi)^3}e^{-E_1/T}$$I don't recognize this notation. How do ...
2
votes
2answers
46 views

How to get exact analytical integral for the basic polarization bubble integral?

In theoretical condensed matter physics, a very common integral that is repeatedly solved is of the following form: $$ \int\limits_{-\infty}^{+\infty}\text{d}\omega\frac{1}{i\omega + i\Omega-a}\frac{1}...
0
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0answers
22 views

How to set PDE coefficient as a list of discrete data

How to set a pde time-dependent coefficient, not as an analytic function in time, but as a discrete data vector or list, each element of that list represents the coefficient value on a specific time ...
0
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1answer
60 views

How to calculate fractional differences of a timeseries?

I have a timeseries I am looking to transform with fractional differences per the following description: The idea being to retain the essence of stationarity of, say, a log transformed integer series ...
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0answers
58 views

One more an incorrect result of FunctionAnalytic

Studying a new in 12.2 command FunctionAnalytic, I execute ...
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0answers
44 views

How to compute an indefinite integral for a general parameter choice?

I am trying to compute the integral: Integrate[E^(a*InverseCDF[NormalDistribution[0, 1], (1 - x)]), x] for a >= 0. The ...
1
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1answer
57 views

Solving a complicated time-dependent function

I want to solve the following time-dependent system of equations: ...
4
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3answers
77 views
1
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2answers
123 views

Why can't Mathematica express a minus operation between two decimal numbers with several places? [closed]

A example of the problem. This is a print from mathematica with the demonstration of the issue.
-1
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1answer
87 views

Asymptotic value of an integral [closed]

When I evaluate the integral $$ n \int_0^1 t^{r-1}(2t-t^2)^{n-1}(2-2t) dt$$ I get the following : $$2^{(-1 + 2 n)}\,n\,t^{(-1 + r)}(\mathsf{Beta}(1/2, n+r-1, n) - 4 (\mathsf{Beta}(1/2, r + n, n))$$ ...

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