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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2answers
38 views

Triple integral: How do I include assumptions and conditions?

I would like to solve the following integral: $$I=\int_{0}^{a} \int_{0}^{y} \int_{0}^{z} e^{(a-x)^{3}} d x d z d y$$ with the condition that: $$a>0$$ So I tried to input the equation like below ...
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0answers
46 views

Is it possible to calculate the given area by an analytical exact expression?

I have the three two-variable curves $f,g,h$ with $0\leq x\leq \frac {\pi}{2} $ and $0 \leq y \leq \pi$ I use ContourPlot as ...
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0answers
53 views

Solving ODE with initial conditions uses a spurious and unnecessary initial value equal to unity

DSolve[ {f'[t] + \[Omega][t] f[t] == 0, f[t0] == 1}, f[t], t] Gives, ...
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3answers
78 views

How to extract a factor from equation and rearrange the equation?

I want to extract a factor from the equations, it's not a common factor, just a factor defined by myself, like this: (ac+b) --> c(a+b/c) I have tried Factor,<...
3
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3answers
190 views

solution of $\frac{1}{x}=\infty$

Is there a way to obtain solution $x=0$ for the equation $\frac{1}{x}=\infty$ in Mathematica? I tried Solve, Reduce, ...
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1answer
74 views

Different results in different versions of Mathematica (computing integrals)

I am working on computing the following integration: aa = N[2, 20] H = 4/160; X = Range[-aa, aa, H]; Integrate[400/(1 + 40*x^2), {x, X[[k]], X[[k + 1]] and ...
4
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1answer
75 views

Substitute known function's derivative

Suppose I know that the time derivative of $x(t)$ is $\dot{x}(t)=-x(t)$. I consider the function $V(x)=x^2$ and I want to compute the time derivative of $V$ along the trajectories of $x$, namely $\dot{...
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0answers
47 views

Closed form for integral with SinIntegral

MMA can't solve this integral: $$\int_0^{\pi } \text{sinc}(x) \text{Si}(2 x) \, dx$$ The I use FourierSinTransform to find closed form,but I try a verify result is ...
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4answers
135 views

Are there any alternatives for NIntegrate to calculate the area for which $f(x,y)<0$?

I have this two-variable function $$f(x,y)= (8 \cos (x+y)+7)\cos \left(\frac{x}{2}\right)+\cos \frac{x-2 y}{2}+2 \cos \left(\frac{3 x}{2}\right) $$ where $0<x,y<\pi$. I want to calculate ...
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0answers
45 views

Can you check analyticity of a complex function in Wolfram Mathematica? [closed]

Can you check analyticity of a complex function in Wolfram Mathematica?
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0answers
30 views

Integration with assumption [closed]

I am trying to evaluate the following integral on Mathematica 12.2 Student Version $ \int_{0}^{\pi} xsin(nx) \,dx $. I tried : ...
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1answer
49 views

Integral with exponential function

Could you please help me in solving the following integration. Thank you in advance. Integrate[((Sqrt[g*(x + h)/y])^V) * Exp[-u * Sqrt[g*(x + h)/y]/n] * Exp[-L*x], {x, 0, Infinity}]
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2answers
220 views

Integrating an exponential with upper incomplete gamma functions

I would greatly appreciate calculating an integral consisting of an upper incomplete gamma function and an exponential function. ...
5
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4answers
168 views

Bilinearization with Mathematica - where to start?

I tried to bilinearize the two equations: ...
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0answers
28 views

Nintegrate of a list of numbers

I have an issue. I have a function in two variables obteined from the table of a numerical integral (done with Nintegrate). Now I have to compute another integral (Nintegral) over it so I will have a ...
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2answers
86 views

Calculating the convergence value of summation

Could you please tell me what value the left side of the inequality converges to? (H and P are constant values greater than 2 and n can go to infinity) $$\sum_{n=1}^...
6
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3answers
305 views

Express MeijerG as integral

For definite integrals MMA gives identities in terms of Meijer G-functions, e.g. $\begin{align}\sqrt{\pi}\int_0^\infty \textrm{e}^{-4x/t^2-t}\ \textrm{d}t &= G_{0,\,3}^{3,\,0} \left( x\left. \...
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1answer
44 views

How to generate assumptions automatically for definite integrals?

In Mathematica, a trick to speed up the process of definite integrals is by adding assumptions. For example, the following codes run faster if we add such assumptions: ...
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0answers
37 views

How can we write code to find the number of patterns except impossible code? [duplicate]

I'm writing a code to find the number of lock patterns on my Android phone. In 3.가능한 패턴의 경우의 수(Number of possible pattern cases), How can we write code to find the number of patterns except impossible ...
5
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2answers
116 views

Efficient computation of Hopf index

I am trying to check using Mathematica the following result: $$-\frac{1}{4\pi^2}\int_{0}^{2\pi}\int_{0}^{2\pi}\int_{0}^{2\pi} \mathbf{F}\cdot\mathbf{A}\;dk_x dk_y dk_z=1,\quad\text{where}$$ $$z_{\...
8
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4answers
271 views

Challenge: Computing the Fourier Series of a List Plot!

I have a bunch of data points (copied from Mathematica below), which I have 'ListPlotted' I would now like to compute the Fourier series of this function (after doing a Fourier transform I assume? Or ...
3
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0answers
81 views

Integrate providing incorrect result for complex exponentials

Consider the integral \[\int_{-\infty}^\infty\frac{(e^{iax}-1)(e^{ibx}-1)}{x^2}\quad a,b\in\mathbb R\] which can be manually computed to be $\pi|\text{sgn}(a)-\text{sgn}(b)|\min(|a|,|b|)$. However ...
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2answers
52 views

Series expansion for two limits of x [closed]

I have a function f($x$) given by the expression $$f (x) = \frac{\left(1+x\left[1-\sqrt{1+x^2}\right]\right)^2-x+x^3\left[1-\sqrt{1+x^2}\right]^2}{1+x^2\left(1-\sqrt{1+x^2}\right)^2}$$ and would like ...
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1answer
79 views

Incorrect results of Integrate

The commands Integrate[1/(z + Exp[I*t]), {t, 0, 2*Pi}, Assumptions -> Abs[z] == 1] (2 \[Pi])/z and ...
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1answer
29 views

How to write a function taking limit of multivariables in sequence?

I'm trying to write a function to calculate the limit of a function on a number of variables in sequence. Here I take a function with four variables x, ...
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1answer
41 views

How to compute the max/min surface area of a donut-shape solid generated by a revolved 2D circle, as the volume of the solid doesn't change?

A donut shape solid is generated by revolving a circle $(x-a)^2+y^2=b^2$ around the y-axis. $a$ is the distance from the center of the hole of the donut to the center of the circle revolved, and $b$ ...
3
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1answer
130 views

Analytical Integration

I have a question on analytical integration. z = y/ps; n = 0.95; A1 = Integrate[((a (a + z) - r^2)/r)^0.95 - (n z) ((a (a + z) - r^2)/r)^-0.05, r] Answer: ...
6
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3answers
308 views

How to rotate a circle around the y-axis and calculate the volume of the 3D solid?

How do I rotate the equation ${(x-5)^2}+{y^2}={3^2}$ around the y-axis and calculate the volume of the plotted 3D solid? I tried the code below but the result doesn't work. ...
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1answer
56 views

How to plot the solutions of $f(z,a)=0$ for $z$ in the complex plane?

I have an equation $f(z,a)=0$ where $z$ is a complex variable $\frac {56}{10}<z<2\pi$ and $a$ is a real variable $0<a<\pi$. I want to plot solutions of this equation on the complex plane $...
1
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0answers
51 views

Why Mathematica is not computing this integral? [closed]

Why is the following integral not computed Integrate[(Exp[-t] - 1)/ Sqrt[2 + Exp[-2 t] - Exp[-t] - Exp[t (-(1/2) - I a) - Exp[t (-(1/2) + I a)]]], {t, 0, 1}]? ...
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1answer
57 views

Complex values for a supposed real integral

I have an integral for which I have checked that it is supposed to be real but there some values of zm which gives a complex value. First I checked the validity ...
-3
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1answer
93 views

Is this even possible [closed]

Is this possible? please provide a step by step explanation
5
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3answers
320 views

How do you find information regarding the derivative?

I have been struggling with an issue that I believe has a simple way to solve. Given that I have extracted a term of a partial differential equation: f(1,2,0) [x,y,z]. In this case, it is similar to ...
0
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2answers
63 views

How to illustrate schematically (animated) solutions of a system of $3$-variable equations by increasing one of the variables with step $0.01$?

Following my previous question here, I am looking an alternative way to solve this problem (the method was not applicable to more complicated situations). I have a system of two 3-variable equations $...
12
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2answers
299 views

Total flux of the gradient of the numerical solution of a PDE through a surface

Trying to solve the following PDE with BC T==1 on a spherical cap of a unit sphere and T==0 at infinity (approximated as ...
5
votes
1answer
129 views

How to plot solutions of a system of equations by changing colours when one of the variables increases by step?

I have a system of two 3-variable equations $SysEqs=\{f(x,y,z)=0\;,\;g(x,y,z)=0\}$ where the domain of variables are $$6 < x < 2 \pi \\ - \frac{15}{100} < y < 0\\ 0 < z < \frac 32$$ ...
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5answers
131 views

How to calculate the total area of $\frac{\cos (2 y) \cos (2 x-y)+\cos (3 y)}{\cos (2 x-y)+\cos (y)}$ enclosed in the range $[0,1]$?

I have this two-variable function $$f(x,y)=\frac{\cos (2 y) \cos (2 x-y)+\cos (3 y)}{\cos (2 x-y)+\cos (y)}$$ where $0<x,y<2\pi$. I would like to calculate numerically the total area of the ...
1
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1answer
37 views

Computing Partial Derivatives of a Function wrt Another Function

I have two functions $P_1,Q_1$ which are both depending on the variables $r_{T_m}, r_{i_f}$, which are theirselves functions of $T_m, mi_f$. I am interested in computing the Jacobian $$\frac{\partial (...
4
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1answer
180 views

Gradient of the numerical solution of a PDE

Solving the following PDE, defined in the same domain of a previous question: ...
1
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1answer
60 views

How to obtain numerical solutions of a system of two equations with two unknowns?

I have two equations with two unknowns $-10<x<0$ and $0<y<15$ $$3 y \cosh (4 x) \cos (y)+(2 x+5) \cosh (x) \sin (y)=0,\\3 x \cosh (x) \cos (y)-(2 x+5) \sinh (x) \cos (y)=0$$ I would like ...
1
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0answers
38 views

Derivative of unknown functions and constants [closed]

I'm trying to do symbolic differentiation with Mathematica for expressions with unkown functions and constans. For example the following expression: $$\text{expr} = \omega_1 Q_1^2 Q_2 + \frac{Q_1^3}{\...
0
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0answers
39 views

Calculating correlational dimension using Mathematica

I am modelling such a system: $T:(x,y)\rightarrow (d+C_2(x\cos\tau - y\sin\tau),C_2(x\sin\tau+y\cos\tau))$, $\tau = C_1-C_3/(1+x^2+y^2), C_1=0.4, C_2=0.9,C_3=0.6,d=0.6$. I am applying $T$ to point $(...
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0answers
41 views

Findroot and arc length,similar situation but does not work

In another problem I was presented with this situation It is to find the arc length of the curve ...
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0answers
62 views

Step-by-step solution for a Limit

Is it possible in Mathematica to get a step-by-step evaluation of some functions? Such that Mathematica is outputting not only the result but all the stages that have led to it? If so, how does one do ...
2
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0answers
53 views

How to find the regions for which the derivative of my function changes sign?

I have an energy function of the angles of the electrons' spins. th1 is vector with (2l+2) elements and each element represents the angle of an individual electron spin. I need to eventually find the ...
2
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2answers
122 views

Efficient way of calculating the arc length of a parameterized curve?

give me the curve :I am asked for the arc length between 0 and 2 (although I think it is between 0 and 2Pi) and it gives as a result 100 ...
1
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0answers
26 views

How to output divergence or gradient of a symbol in the form of partial derivatives? [closed]

As being new to divergence and gradient operators, I am trying to make them more readable, so could anyone tell me how can I output the divergence of u={{ux},{uy},{uz}} in the form of partial ...
2
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2answers
130 views

Two-Points Padé Approximant

My Question is about two points Pade approximant From Mathematica References. I just find the Pade approximant for a real function in one point Example : ...
0
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2answers
29 views

Supremum of a complex periodic signal in symbolic form

For complex periodic signal: $-\frac{8 T \left(2 T \omega \sin (2 t \omega )-e^{-\frac{t}{T}}+\cos (2 t \omega )\right)}{\alpha ^2 \left(4 T^2 \omega ^2+1\right)}$ where $T,\alpha,\omega$ - ...
0
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1answer
38 views

Numerical value of Derivative of Dirichlet L function at some particular points [closed]

I want to find the numerical value for the derivative of Dirichlet L function to compare with an another formula. However I am not able to obtain any numerical values. I tried to use ...

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