# 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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### Finding unit tangent, normal, and binormal vectors for a given r(t)

For my Calc III class, I need to find $T(t), N(t)$, and $B(t)$ for $t=1, 2$, and $-1$, given $r(t)=\{t,t^2,t^3\}$. I've got Mathematica, but I've never used it before and I'm not sure how to coerce ...
14k views

### How can I differentiate numerically?

Mathematica has two ways to integrate: Integrate and NIntegrate. But what about D? ...
6k views

### How to deal with complicated Gaussian integrals in Mathematica?

As we know, for most Gaussian integrals, we can get the analytical result. Now I have many Gaussian integrals to treat, which have the following general form, ...
7k views

### How to find the period of an arbitrary mathematical function?

Is there a function to find the period of an arbitrary (possibly complex) function in Mathematica?
1k views

### Is it possible to write a MMA version of LineInt like Maple?

When I asked this question I found that the available answers relied heavily on the official RegionConvert, but which is very weak at that time. Now, We have a more ...
13k views

### Implementing discrete and continuous Hilbert transforms

What is an efficient and accurate Mathematica implementation of the Hilbert transform, for both continuous and especially discretely sampled functions? This transform relates phase and amplitude in ...
6k views

### How does Mathematica understand branchcuts of the complex logarithm?

Say I have the function $f(x) = x \tanh(\pi x) \log (x^2 +a^2)$ where $a$ is some positive real number. Then it seems to be me that Mathematica when given such a ...
5k views

### Can we teach Mathematica about functional differentiation?

The key relation for functional differentiation is $$\frac{\delta}{\delta f(y)}f(x)=\delta(x-y),$$ where $\delta(x-y)$ is the Dirac delta function, and the usual properties of differentiation (e.g. ...
28k views

### Quick Hessian matrix and gradient calculation?

I am absolutely new to Mathematica and I actually want to try implementing a little optimization method . Long story short assuming I have a predefined two-variable function f(x,y) I want to ...
517 views

### many indefinite integrals do not evaluate in 10.1, looking for the cause

Bug introduced in 10.1 and fixed in 10.2 Many integrals no longer evaluate in V 10.1 when they did in 10.0.2 Here are some 23 integrals as an example, that all produced results in V 10.0.2, but now ...
2k views

### Undocumented use of Integrate: Integrating over regions

I have come across a few questions asking about integrating over regions. And while the answers are impressive there should be a better more consistent way. So my question is, are there ways, ...
2k views

### Symbolic integration error

fixed in 10.1 (windows) I'm running Mathematica 10.0.0 and encountered a disturbing error in the symbolic integration of a rather simple function ...
3k views

### Is it possible to get this 'nicer' solution for an integral from Mathematica?

On a recent CAS-enabled exam question a few weeks ago I was required to evaluate the following integral: $$\int_0^5\left(\sqrt{125-x^3}\right)^2\,dx$$ In Mathematica, using the ...
1k views

Let $c$ and $h$ be real values. I was using Mathematica to compute the limit $(h \rightarrow \infty)$ of the following expression: $$\frac{\left(h^2 +c^2 h^2 + \sqrt{4 h^2+\left(h^2+c^2 h^2\right)^2}... 27 votes 3 answers 7k views ### Visualizing a Complex Vector Field near Poles I've been playing around with a visualization technique for complex functions where one views the function f: \mathbb{C} \rightarrow \mathbb{C} as the vector field f: \mathbb{R^2} \rightarrow \... 27 votes 2 answers 486 views ### Bug in ArcLength? fixed in 10.1 (windows) With Mathematica 10.0.2: ArcLength[Line[{{0, 0}, {1, 0}, {2, 0}}]] ArcLength[Line[{{0}, {1}, {2}}]] (* 2 *) (* 2 *) However, ... 27 votes 1 answer 518 views ### Are greek symbols causing different evaluation? I've updated today to Mathematica 9.0.1.0 from version 8 and found something that absolutely confuses me. Let us define a piecewise function: ... 25 votes 3 answers 583 views ### Backslide of Limit Backslide introduced in 9.0, and persisting through 12.0. A friend of mine showed me this example: Limit[Sum[Sin[Pi*k/n]/(n + 1/k), {k, 1, n}], n -> Infinity] ... 24 votes 5 answers 934 views ### Heuristics, tricks, and hacks in symbolic math Mathematica sometimes fails to compute symbolic solutions when posed in the direct or obvious code, but succeeds when the same fundamental problem is posed in a slightly different way, or when ... 23 votes 5 answers 6k views ### Why this real integral yields imaginary results? This integral yields -1-4Iπ/3 in Mathematica: Integrate[(y - y^2 + x - x^2 + 2*x*y)/(1 - x - y), {x,0,1}, {y, 0, 1}] Since ... 23 votes 3 answers 2k views ### How to efficiently find moments of a multinormal distribution? Update: Starting from V10.0 the build-in Moment is fast enough for practical use. I have a multinormal distribution with covariance matrix \sigma and zero mean. ... 23 votes 1 answer 1k views ### Implement fractional Laplacian What is a way to implement the Fractional Laplacian with Mathematica? How can we apply such implementation to numerically solve the problem$$(-\Delta)^su = 1 \text{ in } B_1(0), \\ u = 0 \text{ in ... 4k views

### Differentiating functions of vectors/matrices?

I'm dealing with derivatives of scalar functions of matrices and wondering if Mathematica can help me here. The standard approach of expanding it in terms of components is cumbersome. As an ...
1k views

### How can the {x,y,z} points that fall on the outer boundary of a set of values be selected and smoothly surfaced?

For a given set of x,y,z values, that may, or may not form a uniform shape, how can the center of the data cloud be found, and the surface points be located and a solid smooth surface created from ...
791 views

### Teaching Mathematica more about DiracDelta and KroneckerDelta

As the documentation and some experimentation indicates, Mathematica contains little information about representations of the DiracDelta and ...
910 views

### Integrate returns imaginary answer for smooth, real function

Bug introduced in 7 or earlier and persisting through 12.0.0.0 or later I'm trying to evaluate the integral: $$\int_0^{\infty} \frac{1}{4 b \sqrt{\pi} r} e^{-(b-r)^2}(e^{4 b r} - 1) \mathrm{d}r$$ ...
2k views

### Incorrect results for elementary integrals when using Integrate

Bug introduced in 8.0 or earlier and persisting through 13.2 or later There is a rather simple integral ($K_0$ is the 0-th order MacDonald function) $$\int_0^\infty e^{-x \cosh\xi}\, d\xi = K_0(x)$$ ...
854 views

### Derivative of a pure function with SlotSequence

I can live with this but I can't figure out why the following is 0: Derivative[f[##] &][x] 0 From documentation for ...
2k views

### Why does Mathematica give the wrong answer when integrating?

Bug introduced in 8.0 or earlier and fixed in 9.0.0 I integrate Integrate[Exp[I Cos[b - c]] Cos[b], {b, 0, 2 Pi}] Mathematica gives: ...
15k views

### How to do implicit differentiation in Mathematica?

I have an equation which is K^(1/2)*L^(1/2)-(K+L) = 24 but I don't know how to do implicit differentiation to find dK/dL because I only know how to do the normal ...
15k views

### How to find the domain and range of a function with Mathematica?

I'm studying calculus and in some exercises I am asked to find the domain and range of a function. Does Mathematica have already a built-in function for this? I can imagine some ways of doing so, ...
10k views

### How can I implement the method of Lagrange multipliers to find constrained extrema?

I want to form the function $h=f-\lambda_{1}g_{1}-\lambda_{2}g_{2}$ where $f$ is the function to optimize subject to the constraints $g_{1}=0$ and $g_{2}=0$ so that I can solve the first partial ...
515 views

### Negative probability?

Bug introduced in 9.0.1 and fixed in 10.0.2 I am trying to get the sum of the squares of seven random variables, all uniformly distributed. This is what I tried. ...
6k views

### How does Mathematica integrate?

Basically, this question can be considered to be an extenstion to my other question. What I wanted to do was this integral as homework (it is indefinite BTW so no approximations using Simpson's Rule ...
1k views

### Solver for COVID-19 epidemic model with the Caputo fractional derivatives

As it is known in biological system with memory it would be suitable to use fractional derivatives to describe evolution of the system. In a current version of Mathematica 12.1 there is no special ...
Context I am interested in doing double contour integral over paths which are defined implicitly. For the sake of debugging, let's assume its $$\oint_{\cal C}\oint_{\cal C} \frac{1}{u\, x} d u d x$$ ...