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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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1answer
10 views

Compare their asymptotic behaviour using the order notation

For each pair of the following functions, compare their asymptotic behaviour using the order notation. f(n) = e^n and g(n)= 3n^2
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Problem with antiderivative which involves DiracDelta

Let us execute in version 12.0 Integrate[DiracDelta[x]*DiracDelta'[x], x] DiracDelta[x]^2/2 and this lasts at least since version 7. However, the ...
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35 views

Z-Transform Solution

I am trying to solve the z-transform of the following sequence using Mathematica: x[n]=-a^n u[-n-1] The known result is $X(z)=z/(z-a)$ with ROC: $|z|<|a|$. ...
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41 views

I need a starter kit for Differential Geometry

In Modern Cosmology, the Geodesic Equation is described as:$$\frac {d^2x'^2}{dt}+\left[ \left( \{\frac{\partial x}{\partial x'}\}^{-1} \right)^l_i \frac{\partial^2 x^i}{\partial x'^j \partial x'^k} \...
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How do I solve the integral over four spherical harmonics?

I want to solve this integral ...
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2answers
305 views

Siginificant speed difference of Integrate between 9.0 and 12.0

I just upgraded from 9.0 to 12.0, and I found some strange difference in timings for the following expressions: a. Integrate[Exp[-a*x^8],{x,-Infinity,Infinity}] b....
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39 views

Derivative operator w.r.t a vector index and change of variables

Description and code lines: Define a scalar product to represent $A_i·A_j$ , $A_i ·B_j$ and $B_i ·B_j$, such that $A_i · A_i$ = 0. This is implemented in the following way: ...
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1answer
39 views

D: Understanding the output of 'n' th derivative of a function involving Exp[1]

I am trying to find the $n^{th$ derivate of a function involving Exp[1] as given below. ...
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1answer
220 views

Why the Kernel crashes on these integrals in V12?

Reported to WRI, [CASE:4331819] This is using V12, on windows 10, 64 bit. Note: these integrals work OK on 11.3 on same PC. Any idea why the Kernel now crashes on these types of integrals? ...
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1answer
60 views

Reduce to Piecewise

There are some similar posts on SE, but none of them seems to exactly address my issue. I need to integrate a rather complicated function (over $x>1$) separately over two domains defined as ...
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1answer
63 views

FindMinimum giving wrong answer

I have a very simple problem that Mathematica is giving a wrong answer to. I want to minimize the following function: $\frac{3+t^2+6\theta +3 \theta^2}{6 t^3}$ in $\theta$ and $t$. Now, plotting the ...
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1answer
59 views

How to substitute by NDSolve solution to plot a new function

Assume I got some function by NDSolve , like y(t) ...
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3answers
306 views

Solve command does not solve this equation!

I tried to solve the following equation with Mathematica: $\left(1-x^2\right) \left(n \left(x^4-2 x^2+5\right)-4 \pi \left(x^2-1\right)\right) \sinh (\pi x) \cosh (n x)+\sinh (n x) \left(\left(1-x^...
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62 views

How to solve this equation in Mathematica?

I wonder what is the best way to solve this equation in MA? $$ y= \frac{\partial}{\partial x} ~ \frac{\partial}{\partial x^*} f $$ where y and ...
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1answer
74 views

How to obtain the symbolic expression for the integral?

I need to evaluate the following expression, $\frac{1}{\Gamma(1-\alpha)} \frac{d}{d x} \int_{0}^{x}(x-t)^{-\alpha}(\exp(\lambda t)) d t, 0<\alpha<1, \lambda\in \mathbb{C}$ using Mathematica to ...
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Efficient application of the fundamental theorem of calculus

How can the following definite integral be computed with Mathematica? $$ -\frac14\int_1^{\infty}\!\!\frac{\sqrt{-4 x^4-4 x^2 z-(y^2+z)^2}}{x (x^2+z)}\theta\!\left(-4 x^4-4 x^2 z-(y^2+z)^2\right)\...
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The most bizarre kernel crash ever. Kernel crash in Integrate under simple different setups. Unable to find cause

Reported to WRI. CASE:4330461 V 12 on windows 10 64 bit. Note: This problem do not show up in V 11.3. Only in V12. For the last 2 hrs, I've been trying to zoom into why V12 kernel crash when ...
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32 views

Solving an integral using assumptions

I am trying to solve the integral Integrate[(Sin(theta))/sqrt((r^2) + (s^2) - 2 rs(cos(theta))), {theta, 0, pi}] with assumptions that ...
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1answer
57 views

Computing the definite integral of a fractional polynomial containing sin(x) and x^n

I need to compute the definite integral defined as $$\int_{-\infty }^{+\infty } \frac{\sin ({x_0}\, \omega )-\sin (x \omega )}{(\sin (x \omega )-\sin ({x_0}\, \omega ))^2+(x-{x_0})^2} \, dx\,.$$ When ...
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1answer
88 views

Real and imaginary parts of complex logarithm

I need to obtain the real and imaginary parts of the below expression with Mathematica: $(- i a - m^2) \ln(\frac{i m^2}{2 a})$, where $a$ and $m$ are real. So we have: ...
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2answers
43 views

Integrating a function with Max function

I'm trying to evaluate the triple integral $$\int_{0}^{1} \int_{0}^{1-x} \int_{0}^{1-\max(x,y)}1 dzdydx$$ in Mathematica. The code I'm using is simply ...
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How can I code Riccati-Bessel functions nd their derivatives?

I tried my best to code Ricatti-Bessel functions and their derivatives, but I am encountering problems. Please guide me on how to code them?
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60 views

Integrate doesn't work correctly in v11.3

This worked in v10.4.1: ...
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1answer
83 views

n-th derivative of an arbitrary polynomial / power series

How can I tell mathematica to give me the symbolic derivative of the following sum $$ \frac{\partial^n}{\partial x^n}\sum_{j=0}^N\gamma_jx^j=\sum_{j=n}^N\frac{j!}{(j-n)!} \gamma_jx^{j-n} $$ I only get ...
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302 views

Applying rules on rules

So I tried taking a total derivative, with constants, but then the ouput had Dt[ ..., Constants -> ...] as an output. I tried to use a rule to remove it from tha output, but it did not work. See below....
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Speeding my code to calculate a density which uses Folner Sequences?

I created a code which calculates $$D(S\cap[a,b])=\lim_{n\to\infty}\frac{\left|S\cap{F_n\cap[a,b]}\right|}{\left|F_n\cap[a,b]\right|}$$ where $D$ is the density of $S\cap[a,b]$ (in $A\cap[a,b]$), $[...
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2answers
64 views

Correctly differentiate wrt product of variables

I have a function f(x) for which I would need to differentiate and then evaluate it to some product x = y*z. Naively, Mathematica does not accept: D[f[y*z], y*z] ...
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0answers
39 views

InverseLaplaceTransform does not evaluate [closed]

I am very new to Mathematica and not a mathematician, so the question may be very trivial. I am trying to find the function $f(t)$ such that $\qquad e^{-e^{s}} = \int_0^{+\infty} e^{-st}f(t)dt$, ...
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1answer
81 views

Is the computation of a certain limit beyond Mathematica?

I am surprised the following computation stumps Mathematica: ...
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0answers
56 views

How can I get continous plot of ArcTan over all quadrants?

Scope: My calculation is an element in the optimization of a mechanism. The geometry is given by following equations. To get angles β and γ without using the inverse function ...
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1answer
85 views

Why should the format of an integral affect the result

This has been reported to Wolfram and confirmed as a bug in Mathematica V12 This follows on from This question I get a different answer for the same integral input in different ways e.g. ...
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2answers
97 views

Incorrect result for integral of series

Here's an integral that comes up in calculating the number of domino tilings of the plane: ...
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3answers
87 views

Intersection of two functions

Could you please try to find the intersections points of this two functions: $ y = x $ and $y=(a+x) \ {\rm e}^ {-2(a+x)} $ with $x\geq0$ Thanks for your help
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Plotting NDSolve function in complex coordinates

I have this system of equations: $- \ddot{z} + \frac{1}{g} \frac{\partial g}{\partial z} \dot{z}^2 + \frac{1}{g} \frac{\partial g}{\partial z^*} \dot{z} \dot{z}^*$ =0 , $- \ddot{z}^*+ \frac{1}{g^*} ...
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1answer
103 views

Result of Convolve is affected by the GenerateCondition

It is very puzzling to observe that the results of Convolve depend on the GenerateCondition option and on the names of variables....
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0answers
44 views

Using FullSimplify to simply expressions involving differential operators

I am trying to simplify some long formulas involving differential operators in $n$ variables. For example, I want to simplify $$\left(\frac{\partial}{\partial x}-\frac{\partial}{\partial y}\right) \...
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0answers
24 views

Establish a certain formula for a constrained integration over an ordered subsection of a 3-simplex

Some ten years ago, I convinced myself that the solution to this problem (although, it would seem, not carried out with this specific command then) ...
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1answer
100 views

Convolution integrals

I want to calculate the following integral $$\int_{-\infty}^{\infty} d \omega f(\omega) g(\omega'\pm\omega)$$ where $f$ has the form $$f(\omega)=\frac{a}{\omega- \omega_0\pm i \eta}$$ and $g$ has a ...
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3answers
75 views

Riemann surface of incomplete gamma function

Michael Trott in his book The Mathematica GuideBook for Symbolics, page 1003, has illustrated a nice way to visualize the Riemann surface of the incomplete gamma function $\Gamma[\alpha, z]$. To plot ...
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37 views

How to solve a system of five second order differential equations with boundary conditions?

I want to solve system of five differential equations of second order with their respective boundary conditions. So, I create a function that depends on their solutions. Such as ...
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0answers
26 views

Multiple integrals in Mma

My question is conceptual: How does mathematica treat multiple integrals? i.e. If it doesn't matter which of the (e.g.) x1,x2,x3 integral I do first, is doing a ...
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1answer
79 views

Plotting a function in 3 dimensions within a domain

In this paper: https://www.staff.science.uu.nl/~beuke106/HypergeometricFunctions/COGP.pdf Any help how to reproduce the plot in Figure:7 It’s the leading order of the complex function, Equation (...
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0answers
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Help with unsolved indefinite integral ! $\int \!r\ln \left( r \right) \sqrt {ar+{b}^{2}+{r}^{2}}\,{\rm d}r$ [migrated]

I really need to solve this indefinite integral: $\int \!r\ln \left( r \right) \sqrt {ar+{b}^{2}+{r}^{2}}\,{\rm d}r$ It seems much more complicated than it looks. I have found a integral table ...
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2answers
63 views

How to shorten the length of the tangent line

I have all the equations down to manipulate the tangent line on the graph... but the tangent line is too long, what would be the best way to shorten it? ...
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1answer
51 views

How can I find the minimum and maximum output of a function and specify the valid ranges and steping of each variable? [closed]

I am a software developer, and it has been far too long since the last time I've needed to do any math this complicated. I've forgotten a lot of the proper terminology, which has made finding answers ...
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0answers
34 views

Integrate Twice over Symbolic Interval [closed]

I would like to solve: $$ \int _0^1\int _y^11dxdy $$ In one function call to Integrate[]. However, evaulating Integrate[1, {x, y, 1}, {y, 0, 1}] (*1-y*) Gives ...
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1answer
97 views

Unexpected behavior from $\lim_{n\to \infty }a^{1/n}$ [closed]

I am computing the limit: $\lim_{n\to \infty }a^{1/n}$ Using the code: Assuming[Element[a, Reals] && a > 0, Limit[a^(1/n), n -> Infinity] ] ...
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0answers
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Moment-generating function of the log-normal distribution. A question about integration

Consider the probability density function $f(x)$ of the log-normal distribution with parameters $-0.1$ and $0.01$: ...
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1answer
51 views

Integral for infinite resistor grid problem

I'm studying the infinite grid resistor problem, and I've noticed that at some point Mathematica solution diverges with the one that's on the theorical study $$\frac{1}{i\pi}\int_{0}^{\pi}\frac{1-e^{...
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3answers
257 views

Computing the surface area and volume of molecules

Here is a somewhat difficult problem I've had for a long while. Space-filling models of molecules can be easily generated by ChemicalData[], ...