# 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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### Strange behaviour of integrals with Cos, Sin, and Exp

Bug introduced in 8.0.4 or earlier and persisting through 11.0.1 During the study of the problem How to solve this integration? I have discovered a strange behaviour of some integrals. I would ...
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### Is there a way to teach integrate new solutions?

I have an integral which I can solve, but integrate cannot: ...
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### Possible Symbolic Integration Bug

Bug introduced between 5 and 8 and persisting through 12.0. I think I may have found a bug, and want to verify is this reproducible in other versions and platforms and not a mistake of mine. I ...
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### Integrating a product of three Spherical Harmonics

The following command is returned unevaluated. The answer is well known to be related to Wigner's 3j Symbol which is also a defined function in Mathematica. ...
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### Help implementing Magnus Expansion

The Magnus expansion is a tool to approximate solutions to first-order linear differential equations (the Wikipedia page is quite instructive and concise) - it's particularly useful because all orders ...
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Consider $\phi:SO(3) \to \mathbb{R}^3$, $R \mapsto (R^\top e_3)\times e_3$ where $R$ is a real $3\times 3$ orthogonal matrix and $e_3 = [0\ 0\ 1]^\top$. Can Mathematica compute the differential of $\... 0answers 104 views ### How to calculate specific area on surface of sphere? I am trying to calculate the solid angle subtended by arbitrary-shaped loops on a sphere's surface. First, I parametrize circular loops by: $$\theta(t,k_{x0},r) = k_{x_0} + r \cos(t);$$ $$\phi(t,k_{... 0answers 102 views ### Trigonometric integral with assumptions fails Consider the following integral Assuming[Element[{n, m}, Integers] && n >= 0, Integrate[Cos[ϕ]^n Exp[I ϕ m], {ϕ, 0, 2 π}]] ... 1answer 294 views ### Implementing positivity constraints over a six-dimensional hypercube This question involves the same subject matter as my previous one (How can one achieve the most accurate estimates of certain six-dimensional integrals under specific constraints?), but with another (... 0answers 140 views ### Total variation integration of a discontinuous function This question derives from this one, about mathematics and Maple. Consider the following discontinuous function: ... 0answers 367 views ### What kind of algorithm does Mathematica use to find limits? Is there any information available about the implementation of the Limit command? In particular, I am interested in how Mathematica computes limits of real ... 0answers 105 views ### Possible bug in Integrate The integral I'm considering is: \int_0^{2\pi}\frac{e^{\pm t\theta}}{4\sinh^2\left(\frac{s+ i\theta}{2}\right)} = 2\pi e^{\mp s}\Theta(\pm s) for s\neq 0 (otherwise there some other subtleties). ... 0answers 202 views ### How to verify the convexity of a function? I have an optimization problem with the following objective function in (x,y)$$ A\log \left(\sum_{i=1}^n x_i\right)+\log\left(1-\frac{f}{n}\left(\sum_{i=1}^n\frac{x_i}{y_i}\right)\right) $$where ... 0answers 90 views ### Huge difference after changing a fraction to decimal I have a limit to calculate. With[{p = 1/2, q = 0.1}, Limit[a^q Integrate[Sin[x]/x^p, {x, a, Infinity}], a -> Infinity]] gives correct result ... 0answers 154 views ### Exploring formal limit definition I found a few demonstrations on the Wolfram Demonstration Project site that help users to explore the formal definition of a limit. That is,$$\lim_{x\to a}f(x)=L$$if and only if for every \epsilon&... 1answer 356 views ### Integral of DiracDelta giving an unusual answer I have been getting a number of seemingly inconsistent solutions to integrals of Dirac delta functions in which the integrand evaluates to DiracDelta at one of ... 0answers 317 views ### Strange behaviour of MMA in derivatives of some standard functions There are some peculiar things to be discovered in derivatives of some standard functions in MMA: Strange behaviour Example 1: Abs We have ... 0answers 1k views ### Symbolic matrix calculus: What's new in Version 9 I have seen some of the related posts, which ask about doing matrix calculation on tensors unknown dimensions. One of the posts mentioned that version 9 has some capability, but it was concerned about ... 0answers 266 views ### Linearised Einstein equations I need to compute the linearised Einstein Equations around a fixed metric$$g_{μν}=\text{Minkowski metric} + h_{μν}$$which is not the flat metric. Does anyone know a Mathematica package or a ... 0answers 550 views ### Spherical harmonic derivative Consider the following substitution Derivative[2, 0][S][th, ph] /. S -> Function[{th, ph}, SphericalHarmonicY[3, 0, th, ph]] which gives correct answer. While ... 0answers 60 views ### Why this change gives different results in Integrate? I'm new in Mathematica and can't understand why changing a number from Real to Integer is giving different results in my equations: ... 0answers 32 views ### Integration$\int dx f(x,y)$by replacing$\int dx f(x,\xi)$for some transcendental number$\xi$Sometimes when I'm doing a hard integral$\int dx f(x,y)$where$y$is a parameter, I replace$y$with, say, the transcendental number$\zeta(\pi)$, Mathematica will evaluate it more easily, and I'll ... 0answers 107 views ### Show a function is decreasing I'm quite new to mathematica, so most of what I know is just the very elementary basics of the language. I'm working on showing that a function is decreasing for$x>0$, but it has two other ... 0answers 82 views ### Derivative of eigenvalues I work with 4x4 Hermitian matrices (r). I want to calculate a derivative of a function f[t,r] (ff[t_,r_]=1/2*D[f[t,r],t]), where the function f depends on the absolute value of the eigenvalues of r. ... 0answers 118 views ### How to test whether an improper integral converges or diverges? To verify the improper integral$\int_{1}^{\infty}x\cdot\left | \sin (x^4)\cdot\sin x \right |dx$converges or diverges,I code ... 0answers 114 views ### Evaluate$\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx\$

Saw this post and noted that it was posted back in 2013. Just tried V11.3 and got no analytic solution. Just wondering if there is a way for Mathematica to give the desired result? Machine VS Human ...