# 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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### Why is my integral with the error function not being evaluated?

I'm new to Wolfram Mathematica and I was trying to evaluate this integral in particular: $$\int_{0}^{x}\frac{e^t \operatorname{erf}(\sqrt{t})\sqrt{\pi t}+1}{\sqrt{\pi t x -\pi t^2}}\mathrm dt$$ The ...
54 views

### Line integral around singular point (Residue theorem)

We have function $\frac{1}{x-\frac{i}{2}}$ which has one singular point $x=\frac{i}{2}$ so if we integrate around it we get by Residue theorem $2 \pi i$. ...
1 vote
220 views

### Evaluating a complicated integral

I am trying to integrate a function Gamma, but the output returns the input. I have the following code: ...
80 views

### How to express the curvature in polar coordinates with DSolveChangeVariables (or without it)?

The curvature k[x] of a plane curve y==y[x] is given by ...
146 views

### IntegrateChangeVariables producing incorrect result

Consider the simple quadratic function: ClearAll[f]; f[x_]:=x^2; Now let's set up the two identical integrals: ...
1 vote
28 views

### Integrate Differential Form with Parameter with DifferentialForms.m?

In this post, it is shown how to integrate a function with parameter(s) and output a function of the parameter(s). I am trying to do something similar with the DifferentialForms.m package. I have ...
21 views

### HomotopyOperator in DifferentialForms.m Not Working

I'm trying to implement and check Faraday's tensor with the DifferentialForms.m package. I code this to implement the Faraday 2-form ...
62 views

### Does IntegrateChangeVariables work with improper integrals?

Trying a new command of version 13.1 IntegrateChangeVariables, I obtain ...
92 views

### Integrate ignoring assumptions

I want to evaluate an integral with all variables real and positive. Specifically, ...
1 vote
67 views

### Error propagation for multivariate polynomial

I am using a simple error propagation of a multivariate polynomial (of x[i_],y[i_],z[i_] ...). The way I am writing is as a symbolic expression and then ...
1 vote
827 views

### How to plot partial derivative of an integral vs. another variable?

Just to take an example, because my original numerical integral was too complex: T[t_] = NIntegrate[Exp[(eta - 1)*t], {eta, 0, 4}] But when I am using ...
927 views

### Evaluation of Derivative in a Module

Can someone help me understand the evaluation procedure in the following two pieces of code? ...
100 views

### Rewriting terms as perfect derivative

I am trying to find conservation laws of a system, and I'd like to rewrite terms of the form $$A\frac{dA^*}{dt}+A^*\frac{dA}{dt}$$ as $$\frac{d}{dt}|A|^2,$$ where $A$ is a complex valued function ...
359 views

### Can Mathematica help a theorem to be alive?

We are used to do limits by means of $\epsilon, \delta$. This is the first step an student encounters limits. Using $\epsilon$ and $\delta$ leads him to be familiar to Logic. I have taught limits and ...
209 views

### Solving an ODE with parameters and taking the limit of the solution

I am very new to Mathematica and already spent a lot of time trying to do this but failed. I am trying to solve an ODE: ...
133 views

### Indefinite integral of x/(a + x ln(x))

My web version of mathematica (student version) can't provide an indefinite integral of $$\frac{x}{a+x\ln(x)}$$ where a is a positive constant. Specifically, I enter ...
137 views

### Possible bug in ResidueSum

Trying a new command of version 13.1 ResidueSum, I obtain FunctionMeromorphic[{(x^2 - x + 2)/x, Abs[x] > 5}, x] ...
202 views

### Behavior of Mathematica 12.3.1, 13.0.1 under certain integrations

I have a couple of modules (Laplace transformation of Plus-distributions) that were completely straightforward to integrate with version 12.0.0. However, recently ...
4k views

### Is there a way to rewrite integrals in Mathematica using u substitution?

In Mathematica, can I give it an integral and a few substitution rules and have it rewrite the integral in terms of those variables?
1 vote
42 views

### Points, 3DPlots and symmetry of Multivariable functions using Mathematica

My doubts here are related to the construction of 3D plots of Multivariable functions. As a consequence of that, I am going to keep these questions in this single post rather than creating several ...
27 views

### Taylor series two variables and CoefficientList [closed]

I have the code ...
1 vote
71 views

### How to simplify a formula with integrations by parts?

I have a huge expression to compute, which contains lots of different terms composed of multiplication of various functions, derived a certain number of times (non derived, derived once or twice). For ...
2k views

### Change integration variable

Are there any automated procedures to change integration variables? For example by changing Sin[u]->x, I can change ...
4k views

### How to change coordinates of integration?

Is there some built-in routine, some easier method to change variables of integration, without/before solving the integral? Say I have ...
1 vote
77 views

### Weird Infinite Limit results when dividing functions

I am getting 0 or 1 for the limit when dividing the two functions but Log of the same constant when taking the limit individually. ...
39 views

### Get different results when evaluating indefinite integral multiple times

I'm trying to evaluate the following indefinite integral in Mathematica: Integrate[Exp[(k + g) t]/(1 + Exp[k t])^3, t] When evaluating the integral multiple times ...
57 views

556 views

### Defining the Hirota operator

The Hirota $D$-operator (derivative) is mathematically defined as follows: $$D_x^n f\cdot g=\left.\frac{\partial^n}{\partial s ^n} f(x+s)g(x-s)\right|_{s=0}$$ An example of this operator acting on ...
52 views

### Real valued solution

I want to obtain the critical points or if you like a real solution for the first derivative of the function set to zero, and the second derivative also vanishes of the function : ...
180 views

### Mathematica giving wrong answer for limit of LerchPhi[]

Mathematica is often frustrating to use, due to how it apparently get things wrong at times. It's giving me a totally wrong answer for this below limit (it gets it right though, if I pick a positive ...
1k views

### Computing Christoffel symbols of the first kind

I am trying to compute the Christoffel symbols of the first kind. To keep the question short I am only computing the second term: $$\Gamma _{\text{ijk}}=\partial \text{Mg}_{\text{ik}}/\partial q_j$$ ...
37 views

### Integer parameter assumption gives incorrect result when integrating

There are some dangerous pitfalls that can happen when evaluating some integrals, assuming that some parameters are integers. For example, consider the following: ...
1 vote
107 views

### How can I simplify the output after integration which is obtained in terms of hypergeometric function in Mathematica?

When I did the integration of the following function, I got the answer in terms of hypergeometric function. But when I did the same integration in MATLAB, a simplified form of expression is coming as ...
1 vote
121 views

### Simplification of Elliptic integrals

Consider an indefinite "elliptic integral”, $$J_n(r):= \int^r_{r_0}\frac{x^n {\rm d}x}{\sqrt{f(x)}}, \quad f(x):=(x-x_1)(x-x_2)(x-x_3),$$ Here, $r_0>0, n=1,0,-1$, and $x_{1,2,3}$ are roots ...
43 views

### Estimating a limit of a sequence by several values

Given first several values of a sequence, e.g., {0.291088365036517,0.34379330139259,0.370496101167372,0.383547058928976,0.389873503208829} I'd like to to find the ...