# 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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### How to find the local maximum and minimum of a function

I am trying to find the local maximum and minimum of a function. The code below works if function is an exponential but when function involves a sinusoid, it does not give numerical values and instead ...
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### How to help DSolve obtain this solution to $y'(x)=\sec (y(x))$

V 14 ode = Cos[y[x]]*y'[x] == 1 ic = y[0] == 2 DSolve[{ode, ic}, y[x], x] Anything can be done to help DSolve obtain the following solution? Hand solution \begin{...
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### Function returning error every other run

Disclaimer: I am new to Mathematica but have some experience with other programming languages. I have defined the following function (simplified version): ...
1 vote
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### Why does functional form of integration output depend on variables used?

The following integration code is identical, except for the use of integration variables x vs k and constants ...
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### How to plot the contour for temperature in a wedge surface?

Through conformal mapping, the temperature distribution $T$ on a wedge sector can be obtained. With Mathematica, how to display the distribution? $$T(x,y)=\frac{T_0}{\theta_0}\arctan\frac{y}{x}.$$ ...
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### Eliminating complex numbers from solutions

In Maple eq:=ln(y)-1/2*ln(1+y^2) = x-c1; solve(eq,y) In Mathematica ...
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### Errors in NIntegrate - or in coding?

As far as I know, if a set of functions f,g,h,... are all Riemann integrable then the definite integral of the sum f+g+h over ...
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### How to calculate the PDF of product of two random variables from generalized gamma distributions?

Namely, we want to find the explicit formula of PDF of double Generalized Gamma distribution. ...
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### Euler-Maclaurin method of integration

I am trying to calculate the following integral using the Euler-Maclaurin formula. However, when 𝑎a and 𝑏 are not integers but real numbers, defining the sum, for example, between 1/3 and 1/2, doesn'...
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### How to derive a closed-form expression for the sum of this series?

Mathematica 14.0 on Windows produces Sum[x^(9 + 6 k)/(9 + 6 k)!, {k, 0, Infinity}] ...
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### Integral of $1/\sin(y)$ in Wolfram Mathematica vs. Wolfram|Alpha

I was looking at solution of this ode using step-by-step in Wolfram|Alpha: But in Mathematica V 14 Integrate[Csc[y], y] gives ...
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### Integrate regions in Plot/Piecewise

I have the following code: ...
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### Evaluating Integral

I am having trouble evaluating the following integral, which they say has a solution! Thanks. ...
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### Numerical Simulation of a Damped, Driven Nonlinear Wave System with Spatially Extended Initial Conditions

I am working on a project that requires creating a specific type of graph, but I am having trouble writing the correct code. The graph should look similar to the one I have attached below. Could ...
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### How to generate lattice by integrating over curves on a modular form domain plot?

I was watching this video (2:30 of https://youtu.be/zLEyIT_BCgk?si=ji5NyjR7vcwzaNxi ). Here in the video he said that to get the lattice , we just integrate over the curve(silver line) and we will ...
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### Different result integrating with integer bounds vs floating point bounds

Someone posted this question at community https://community.wolfram.com/groups/-/m/t/3178350 Since little attention is paid for it there, I thought here will get more. ...
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### Unable to solve an integration

I would like to solve the following integration: denom=44.3705 - 17.0692 Pe + 17.0692 Pp - 38.3886 Pe Pp + 35.6879 x - 13.1912 Pe x + 13.1912 Pp x - 43.2119 Pe Pp x + 22.5263 x^2 - 6.59222 Pe x^2 + 6....
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### FunctionConvexity of power function $x^a$

I'm trying to check convexity of some functions. I encoundered a problem with the line ...
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### Separation of variables

Is it possible to replicate the same procedure of placing the functions that depend on $r$ on one side of the equation and on the other side leave the equations that depend on $\theta$ in Mathematica? ...
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### IntegrateChangeVariables sometimes doesn't work

For example, the following simple code doesn't work: IntegrateChangeVariables[ Inactive[Integrate][Sqrt[(1 + x)/(1 - x)] , {x, -1, 1}], t, (1 + x)/( 1 - x) == t] ...
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### Kernel quits while using ParametricNDSolve

I'm trying to reproduce the plots (Figure 1 and Figure 2) of this paper. The plots are obtained by solving the Eqs. 4.3 and 4.4. The boundary conditions described in section 3.3 for Eqn. 4.3 are ...
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### NIntegrate with depended variables

I have problems in integrating a function with depended variables. I am pasting the code below ...
1 vote
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### Find the polar plane of the ellipsoid and integrate the Ellipsoid cap

I am working on writing a code to find the polar plane of an ellipse, and subsequently calculate the cap formed by subtending the point $p$ (red) onto a random ellipse. I think I have managed to ...
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### Drawing cycles around and across cuts

I want to draw cycles (or paths) around and across the cuts on the complex plane for the following function (where I have put the cuts between [z1,z2] and ...
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### Why does singular solution not satisfy the ode?

Should not singular solution also satisfy the ode itself? Since it is a solution. Why does Mathematica V 14 in this example generate a singular solution which does not seem to satisfy the ode itself <...
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### Symbolic integral with piecewise functions

I want to compute the following integral as a function of some positive real parameters, $a, \rho, \lambda$ and three positive real variables $L_a, L_b, L_c$: ...
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### Is it possible to find a closed-form expression for determinant of the $k+1$-dimensional three-diagonal matrix whose entries obey particular rules?

I have this $k+1$ dimensional three-diagonal matrix in which $k=0,1,2,3,...$. I want to compute its determinant. Of course, for each fixed $k$ it is easy to compute the determinant. My question. Since ...
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### Integrate takes long time

This Integration does not return for hours, though I don't know why as the graph doesn't look all that complicated. Also NIntegrate returns the result instantly... but I need the exact value! ...
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### Two-dimensional integral over data

I am trying to calculate the integral of the data. Following this post, I know what to do with a one-dimensional list. But, let's consider a simple function, just to show you, what I want to do. ...
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### How to calculate a definite integral with a parameter

I am trying to calculate the following definite integral with the parameter H. So my result would like to be a analytical one with the parameter H. For example for the integral NIntegrate[ax, {x, 1, ...
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### How to obtain the boundary of a two-dimensional ParametricRegion surface in order to apply Stokes' theorem to compute surface integrals?

On the help page of the SurfaceIntegrate, an example of Stokes' theorem is listed: Compute the Curl ...
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### Contour integral with radius infinite [closed]

How can I get compute a contour integral of the function $f(z)=\exp(i z)/z$ in the upper circle with center in $(0,0)$ and radius $R$? I want verify that $\lim_{R\to \infty}\int_C f(x)=0$ in ...
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### Issue with FullSimplify in StirlingS2 definition

I have previously asked a question about a problem simplifying an expression in Mathematica. The solution was to expand the definition of StirlingS2. This time, I ...
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### Problem calculating mean of probability distribution

I have a discrete probability distribution where $P(X=x)=(Factorial[n]/(n^x))*StirlingS2[x-1,n-1]$. So, when trying to calculate its mean with the following property: $\mu = \sum_{i} x_i \cdot P(x_i)$,...
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### Parabola from a focus and a directrix

The code below is an attempt to obtain the graph of a parabola from two pieces of information: a focus and a directrix. I am not able to get the correct graphical shape of the parabola. I appreciate ...
1 vote
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### How can I ask Mathematica to give the explicit real form of the given function?

In part of my calculations, I obtain this expression which contains the imaginary unit I but I expect that this expression might be real (from the comments below, ...
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### Questions About the FunctionExpand Function

Today, while helping someone solve a problem, I encountered an issue with the Abs function in their expression. I needed to derive the ...
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### Taxicab Geometry

According to taxicab geometry, there are 1226 possible paths from A(2,5) to B(7,9), all with a distance of 9 units, |2-7|+|9-5|=9. I wanted to write a code where I could be obtaining the plots of some ...
I'm trying to find the maximum $B$ of the following function: ...