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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46 views

Find analytic solution for integral only defined for even integers

I would like to calculate the integral $$\int_0^{2\pi} dx \sin^6\left(\frac{x}{2}\right) F\left(\frac{4-n}{2}, \frac{4+n}{2}, \frac{1}{2}, \cos^2 \frac{x}{2} \right)^2$$ where $F$ is the ...
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2answers
47 views

Double integration with a piecewise function gives wrong result

I am attempting to solve a double integral $$I(t) = \int_0^t dt' \int_0^{t'} dt'' f(t'')$$ of a piecewise function of the type $$f(t) = \begin{cases}1 & \text{for } 0 \leq t < a \\ -1 & \...
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21 views

Multiply Taylor expanded matrices with multiple variables and different order

I want to multiply rotation matrices (Euler matrices) that are expanded in Taylor series to a different order. When I use Series on the separate matrix, the multi-...
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39 views

Chain rule applied to a kinematic problem

I have a very simple question. How can I enter and solve the following equation in Mathematica? thank you for your help dv/dt = ds/dt dv/ds A supplementary question on the same subject. What is the ...
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1answer
50 views

Plot Point On StreamPlot

How would I go about plotting a single point in the $x,y$ plane given by; $$\left(\frac{\lambda d -b\mu}{ad-bc}, \frac{a \mu -c\lambda}{ad-bc} \right)$$ Where we can take $a,b,c,d,\mu$ to be positive ...
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45 views

Solving PDE IVP with inequalities in domains

I want to solve the following problem in Mathematica: I used the following Mathematica code. I am unsure whether this is the correct way of entering the assumptions for the initial and boundary ...
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37 views

Plot inside Streamplot [duplicate]

I am looking to include plots in the $x,y$ plane that represent the lines that the fixed points of a dynamical system follow as the variable $l$ is varied. In doing this, I am trying to represent ...
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2answers
80 views

Fitting plot and data to an equation

If I have the following data: ...
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9 views

The convergence/divergence of lim x>0 ln(1+f(x))/f(x) when f(x)=sin(1/x)x^2 [migrated]

This question came up between me and my teacher when we discussed whether the following when lim x>a f(x) = o does lim x>a ln(1+f(x))/f(x) = 1 is true or not. It is widely known that lim x>0 ln(1+...
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3answers
139 views

On correct use of Limit for multivariable function [closed]

V 12.1 on windows. This limit $\lim_{(x \rightarrow 0,y\rightarrow 0)} \frac{x^2-y^2}{x^2+y^2}$ depends on the direction. So the limit does not exist, or could be written as Maple does it, which is $...
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48 views

Is zero multiplied by infinity indeterminate or zero?

Is $0\times\infty$ indeterminate or zero? Mathematica says the result is zero. Limit[0*n, n -> \[Infinity] ] or ...
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1answer
78 views

Can you force Integrate[] to find a complete symbolic solution for all variables?

(As I've asked on the Math StackExchange and on a related previous question), I am interested in getting a complete symbolic solution to the integral of an expression with a lot of unassigned ...
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1answer
87 views

Results from Integrate not matching NIntegrate

I am interested in getting a completely-symbolic form of an integral of an expression. But I noticed that when I compare my symbolic integral with a numerically calculated integral, I see that there ...
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1answer
186 views

Changing the integration limits takes a long time to evaluate

Integrate[x*Sin[x]/(1 + Cos[x]^2), {x, 0, π}] π^2/4 But ...
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2answers
146 views

Finding simplifying substitutions for an integral involving limits and integrand

[The following is based on a William Lowell Putnam Mathematical Competition problem.] Consider the definite integral: $I = \int\limits_2^4 \frac{\sqrt{\log (9-x)}}{\sqrt{\log (9-x)}+\sqrt{\log (x+3)}...
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63 views

Convective derivative in cylindrical coordinate

My question is similar to this one about convective derivative. But how do I adapt it to the cylindrical coordinate? Simply changing Grad to cylindrical coordinate ...
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2answers
763 views

The time-like geodesics (orbits) in the Schwarzschild spacetime

I am trying to plot Schwarzschild's orbit without invoking the geodesic equation. As a reference I am using Chandrasekhar's Book (The Mathematical theory of Black Holes, Oxford University Press). In ...
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2answers
54 views

Using different baselines to estimate area of two peaks

I have the following data: ...
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1answer
103 views

Why can't Mathematica integrate this integral?

I'm trying to estimate this integrate: Here Θ is the Heaviside theta function. And I can use PR[k_] := 2.1*10^-9 (k/0.05)^.04 I want to solve this integral for ...
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1answer
73 views

Unclear result from Leibniz Integral Rule

I was trying to answer this question: $\frac{\mathrm d}{\mathrm dt}\int_{t-\mathrm d1}^t \int_h^t f(s) \,\mathrm ds\,\mathrm dh$. I got the accepted answer by hand (using the Leibniz integral rule) ...
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1answer
62 views

Mathematica integrates centered functions, but can not integrate shifted ones

Mathematica seems to integrate this function: $\int \limits_{-\infty}^{\infty} d w\, \frac{\sin ^2\left(\frac{1}{2} wt \right)}{w^2} \frac{\frac{\gamma ^2}{4}}{ \left(w^2+\frac{\gamma ^2}{4}\right)}$, ...
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35 views

Derivative for conditional expectation of non-independent gaussian variables is increasing

Consider two non-independent gaussian random variables: $$(t,c)∼\text{BiNormal}[(𝜇𝑡,𝜇𝑐),(𝜎𝑡,𝜎𝑐),𝜌]$$ I'm interested in understanding when (i.e. for which values of the distribution ...
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63 views
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43 views

Understand output of Integration

Could someone help me understand the output of the following Integration in Mathematica? Specifically, (a) what is the meaning of "True"? (b) Why is the answer in two parts which seem to overlap ...
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1answer
65 views

Define function of n variables and integrate [closed]

Im looking at n particles and functions of their coordinates which I want to integrate over: ...
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1answer
40 views

Derive arbitrary steps between two expressions

Can someone show me the syntax to find the algebraic transformations between: $\sqrt{1+(\frac{dy}{dx})^2}dx$ and $\sqrt{dx^2+dy^2}$ And more generally, how to get the algebraic derivation between ...
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0answers
90 views

Rubi/Mathematica gives different values for the same integrand

Consider this integral: $$ \int_0^\pi d\theta \sin\theta\, {1 \over \gamma^q}\left((1-\gamma^2)^2\sqrt{1 \over 1-\left(1-(1-\gamma^2)^4\right)\cos^2\theta} - 1\right)^q $$ for $0\leq \gamma < 1$. ...
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1answer
68 views

complex number from the integration of a real integrand

Consider this integral: $$ \int_0^\pi d\theta \sin\theta\, {1 \over \gamma^q}\left((1-\gamma^2)^2\sqrt{1 \over 1-\left(1-(1-\gamma^2)^4\right)\cos^2\theta} - 1\right)^q $$ for $0\leq \gamma < 1$. ...
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1answer
58 views

Two different Integration results for the same integrand in Mathematica

I want to calculate $$ \int_0^\pi d\theta\sin\theta\, \left(\sqrt{1 \over 1-\left(1-(1-\gamma^2)^4\right)\cos^2\theta} - 1\right)^q $$. For the simplest case of $q=1$ we have ...
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1answer
114 views

Integration not working

Can't seem to get this integration to work. s, n, b, b, l are all constants. ...
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47 views

Can anyone help me to get symbolic expression for the following code?

Using Mathematica I did some manipulation on the function $\sum _{s=-m}^{m} e^i \lambda ^{-(2-d) s} \left(1-e^{i y \lambda ^s}\right)$. And got the following output: (1) when I took $\text{m}$ in ...
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0answers
62 views

Faster Way of Plotting a Multivariate Integral with No Closed Form and Maximized

I have a double integral that looks like this. $$J=\int_0^2\int_{-2}^2 \sin(y\cdot B_z+z\cdot B_y+\phi_y+\phi_0)\,dy\,dz$$ $\phi_y$ I am defining as different functions of y, e.g. $\phi_y=0$, $\...
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1answer
86 views

Jacobian Matrix?

I am doing r[psi_, th_, phi_] := {Cos[(phi + psi)/2]*Cos[th/2],Cos[(phi - psi)/2]*Sin[th/2], Sin[(phi - psi)/2]*Sin[th/2], Sin[(phi + psi)/2]*Cos[th/2]} and ...
2
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1answer
107 views

Line integration given tangent vector

I have the following problem: I know a unit vector in spherical polar coordinates, let's call it $\vec{e}_{s}$, in the form $\vec{e}_{s} = f(r,\theta)\vec{e}_{r} + g(r,\theta)\vec{e}_{\theta} + c\vec{...
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1answer
31 views

How to get the output for the following code involving differentiation and integration?

I need to evaluate the following Mathematica code to get a numerical value as my output. But during compilation I get the error message NumericalMath`FixedPrecisionEvaluate::precbd: Requested ...
2
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4answers
261 views

Series expression of a RootSum object

Mathematica was not able to calculate the definite integral of a trigonometric function of mine: ...
1
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1answer
64 views

Fractional Euler-Lagrange equation [closed]

Using Mathematica, I need to perform the derivation of the fractional Euler-Lagrange equation as derived in this article. Regarding this, I searched many links, but I was not able to master the ...
3
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3answers
211 views

How to calculate arbitrary 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_{...
4
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2answers
341 views

Why doesn't Integrate evaluate an elliptic integral?

My code is Integrate[ 1/Sqrt[(x - a) (x - b) (x - c) (x - d)], {x, a, ∞}, Assumptions -> 0 < d < c < b < a] I know this can be ...
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2answers
159 views

Why is my integral not being evaluated?

I am trying to integrate a function Gamma, but the output returns the input. I have the following code: ...
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1answer
68 views

Error in Taylor's Series using Mathematica

How many terms should be used in taylor series expansion of the function f(z) = e^z around z = 0 for a specific value of z = 30 + 30 i to get an error of less than 0.05 using Mathematica? ...
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3answers
385 views

Calculating curvature of a contour

I have an equation of a scalar field in the form $$f(x, y) = x^2 + y^2 + xy + c$$ I want to find the curvature of the contour of the curve at $f_c = f(0.5, 0.5)$. So I need to calculate the ...
2
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0answers
50 views

Why Integrate does not obtain this conditional result automatically for orthogonality of cosines?

V 12.1 on windows. I remember asking something similar many years ago. I was hoping Mathematica now could have done this automatically: $$ \int_{-\pi}^{\pi} \cos (n x) \cos (m x) \, dx $$ For ...
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0answers
63 views

Why doesn't Mathematica simplify? [closed]

Consider the following: Simplify[Integrate[x^(a), {x, x1, x2}], x1 > 0 && x2 > 0 && a > 0] Mathematica doesn't simplify this for me. I ...
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1answer
4k views

Kramers-Kronig relations

I am trying to calculate the change of the refractive index from the change of the absorption coefficient using the Kramers-Kronig relations, in Mathematica. ...
2
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1answer
52 views

Plot the expression using NIntegrate for the limiting case

I need to plot the below equations: $W^{+}(x,\tau,\alpha)]=-\frac{2m}{\pi\alpha}\exp[{\alpha x}]\times\int_{0}^{\infty}\frac{G(\omega,\tau_c)}{1+\omega^2}\cos(2\tau_c\omega\sqrt{1+\omega^2}-2\tan^{-1}...
2
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2answers
228 views

On evaluating an integral related to Chern numbers

I am trying to evaluate the following integral in Mathematica $$ ∫_{-\infty}^{\infty}\mathrm dk_1∫_{-\infty}^{\infty}\mathrm dk_2 \frac{mπ}{(k_1^2+k_2^2+m^2)^{3/2}}\frac{Δ(k_1^2-k_2^2)}{\sqrt{[Δ(k_1^...
5
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2answers
730 views

Bug in Integrate?

In v. 11.1.0.0 Mathematica for Linux Integrate[Sin[Cosh[t]], {t, 0, Infinity}] returns Integrate::idiv: Integral of Sin[Cosh[t]] does not converge on {0,[...
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0answers
6k views

What's the most difficult multidimensional integral that Mathematica has solved?

I am interested in benchmarking numerical integration methods and am trying to develop a wider set of difficult multivariate examples. For my particular methods, I only want to look at non-negative-...

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