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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Coupled heat transfer equations using collocation method

Using the collocation method proposed here, recently this problem has been solved. I am trying to solve a similar problem described by the equations given below. My attempt in Mathematica is ...
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1 vote
1 answer
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How to get this expression in a wanted form

This is the endresult of integration calculation (I*Log[(b - I*z)/(b + I*z)])/(2*b) See the question integral calculation post This endresult must transformed to a ...
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Evaluation of this definite complex integral

I want to obtain the symbolic expression for this definite complex integral: $\int_0^z \frac{1}{b^2+z^2} \, dz$ Then doing a subtition z in the Integral's upper boundary ...
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Integration shows different result after manual u-substitution

I'm working with sine and cosine integral functions, and stumbled on this weird issue when performing integration in Mathematica. I worked out the following integral by hand and found: \begin{equation}...
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Is this a bug in Integrate?

I try to find with Mathematica 13.0.0 on Windows 10 the following integral (This is not art for art's sake, see McCrea W. H., Whipple F. J. W.Random paths in two and three dimensions, Proc. Roy. Soc. ...
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4 votes
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Finding Coefficients in a Perturbation Problem related to Chaotic Dynamics

I am trying to reproduce the results from this paper: Chaotic dynamics of a suspended string in a gravitational background with magnetic field I am currently stuck with finding the coefficients. First ...
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3 votes
2 answers
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Roots, multivariable functions and Mathematica [closed]

Let $f(x,y)=(10 x^2 + 4 x y - 2 x + 4 y^2 - 4 y + 1)^2 (32 x^2 - 64 x y + 24 x + 40 y^2 - 28 y + 5)^2$ ...
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13 votes
2 answers
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why these integrals fail to evaluate in V 13 compared to V 12.3?

This was also reported to WRI. [CASE:4903465] I'd like to know why these integrals no longer evaluate in V 13 and timeout after 3 minutes while using the same PC and OS they all evaluate quickly in ...
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1 answer
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Equating the coefficients to get following relation

I have a system of differential equations as follows: ...
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1 answer
114 views

I'm beginner and do not how to plot the Basins of attraction by the methods

For a complex polynomial $P(z)$, define $T$ by $T(z)=z-\frac{P(z)}{P^{\prime}(z)}$, where $P^{\prime}(z)$ is the first derivative of $P(z)$. We consider Newton method for finding the roots of ...
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1 vote
2 answers
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How to get the implicit function solution of ODE?

For example, ODE: $$(2x - 5y + 3)dx - (2x + 4y - 6)dy = 0$$ The implicit function solution: $$(2x+y-3)^2 (4y-x-3) = C$$ However, if it is solved directly with DSolve, the solution is very complex: <...
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How can I perform this calculation in another program done, also in Mathematica? [closed]

This is done in Maple in steps (see picture) , and i want to do this on the same manner with steps in Mathematica.
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1 answer
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How do I get the n-th derivative of the function f(s) [closed]

How do I get the n-th derivative of the function f(s) The result of the n-th derivative is also shown in the picture and the n-th derative
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3 votes
1 answer
115 views

How to generate a 2D region from a 1D region boundary?

For example: ...
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2 votes
1 answer
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How to improve the resolution of Region image boundary or the density of boundary sampling points?

I want to draw the region between two function curves (Sqrt[x], x^2): ...
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5 votes
1 answer
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How to get the expression of a special function?

How to get the expression of a special function? Like gamma function: Gamma[z]=Integrate[t^(z-1)/e^t, {t, 0, Infinity}]
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-1 votes
1 answer
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How to find then-th derative for function f(s) [closed]

How to calculate the n-th derative for f(s)
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2 answers
148 views

Convergence of a complete elliptic integral

How to prove that the following integral assuming $k^2 < 1$ $$\int_{0}^{1} \frac{d x}{\sqrt{1-x^2}\sqrt{1-k^2 x^2}}$$ is convergent? ...
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0 votes
1 answer
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Weird Infinite Limit results when dividing functions

I am getting 0 or 1 for the limit when dividing the two functions but Log of the same constant when taking the limit individually. ...
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1 vote
0 answers
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Numerical Integration and Fractional Decomposition (Apart) [closed]

Suppose we want to perform the following integral: $$I=2\int_{-1}^{1}f(e_0,s)g(e_0,s) \mathrm{d}s$$ with: $$f(e_0,s)=\frac{16 s^2 \left(2 s^2-1\right) \left(8 s^4-8 s^2+1\right) \left(8 s^2 \left(8 s^...
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1 vote
1 answer
160 views

How to let Mathematica return impulse or Dirac delta functions when computing integrals?

For example, let's say I want to compute the (continuous-time) Fourier transform of the signal/function $\cos{(3t)}$, which is given by the following improper integral: $\displaystyle\int_{-\infty}^{\...
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2 votes
0 answers
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Possible bug in Integrate: convergent integral reported divergent

$Version "8.0 for Microsoft Windows (64-bit) (October 7, 2011)" on Windows 10 Consider the calculation of this integral ...
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3 votes
2 answers
124 views

Double integral over special region [duplicate]

I need to calculate the value of the integral Over the region between the four lines I thought I could perhaps define the region using ...
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  • 127
3 votes
1 answer
96 views

Can I plot a function in terms of its derivatives?

I have sign conditions on the partial derivatives of a function of two variables $f(x,y)$ and its cross-derivative, but I do not know the function itself. Can I visualise a prototype of such function?
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0 answers
42 views

Are these 3 divergent integrals regularization methods equivalent?

I implemented in Mathematica 3 methods for regularizing divergent integrals, and wonder if they are equivalent. Code: ...
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0 votes
2 answers
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Mathematica and analytical solution not matching

I have the function $$f(r,t)=r^{2n}-2 r^n\cos nt+1$$ where $n $ is an integer. I want to find the maximum value of $f$ over $(r,t) \in [0,1] \times[0,2 \pi]$. As usual in order to find the critical ...
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5 votes
1 answer
151 views

Multiple Integration

I tried the following integration $$\frac{1}{(2 \pi \hbar)^3} \int_{-\infty}^{\infty} dp^3 exp[i\frac{\vec{p}\vec{r}}{\hbar}-i\frac{ip^2t}{2m\hbar}]$$ with this code but failed to have the right ...
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  • 107
0 votes
2 answers
52 views

Integrate command to calculate definite integral [closed]

I'm trying to evaluate the following expression: where 0 < x 0 y. I'm using the following command, but can't get it to return a result. ...
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1 vote
0 answers
91 views

Using Runge-Kutta to solve Black Scholes [duplicate]

I am given the Black Scholes equation as $$\frac{\partial c(t, s)}{\partial t}+\frac{1}{2} s^2 \sigma^2 \frac{\partial ^2c(t, s)} {\partial s^2} + r s\frac{\partial c(t, s)}{\partial s} − rc(t, s) = 0 ...
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2 votes
1 answer
100 views

How to evaluate conditional expression for definite integral [closed]

I am trying to show that the definite integral Integrate[ (y*(1 - y) + (1 - t)*y^2 + t*(y - 1)^2)/((1 - t)*y^2 + t*(y - 1)^2)^2, {y, -Infinity, Infinity} ] is $...
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1 vote
0 answers
58 views

Finding the maximum and minimum of a function two variables involving a parameter [closed]

I have the function $$f(r,t)=r^{2n}-2 r^n\cos nt+1$$ where $n $ is an integer .I have the function $$f(r,t)=r^{2n}-2 r^n\cos nt+1$$ where $n $ is an integer .I want to find the maximum value of $f$ ...
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2 votes
1 answer
63 views

Identical integral evaluating differently

I am trying to compute an integral but am getting two different answers using two near-identical methods. I wish to know why they differ. ...
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10 votes
4 answers
568 views

Integrate returns "diverging integral" for obviously convergent expression

Integrate[Cos[phi]^2*Sin[phi]^n, {phi, 0, Pi}, Assumptions -> n > 0] on Mathematica 13.0 returns the obviously mistaken error message (...
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5 votes
1 answer
240 views

What integrals can `FourierTransform` evaluate that `Integrate` cannot?

FourierTransform can make sense of integrals that diverge according to Integrate. ...
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15 votes
1 answer
276 views

Integration Bug?

Bug introduced in 8 and persisting through 13.0 Mathematica 11.2.0.0 Home edition on MacBook Pro OS X 10.10.5 (Yosemite). ...
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3 votes
2 answers
126 views

How to integrate a symbolic sum?

I'm trying to integrate a function that involves a finite sum: $$\int_{-\infty}^{\infty}\sum_{j=1}^n (e^{-b t^2}r_j) \,dt$$ I think it should be possible to take the exponent outside the sum: $$\int_{-...
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6 votes
1 answer
207 views

Using Integrate and then Series seem to produce a wrong result

Run this: ...
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2 votes
2 answers
156 views

Derivative of a definite integral

The derivation process of a definite integral is as follows: (* D[Integrate[(1 + t^4)^(-1/2), {t, x^2, x^3}],x] = (3x^2)/Sqrt[1 + x^12]-(2x)/Sqrt[1 + x^8] *) However, the results obtained by ...
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3 votes
1 answer
111 views

Change of variables in 2-nd order PDE

I need to solve the following equation: $-(1+x) \partial_t ^2 \psi + (1-2x^2)\partial_x\partial_t \psi +(1-x)x^2\partial_x^2 \psi -2x \partial_t\psi+x(2-3x)\partial_x\psi=0$ for the function $\psi(t,x)...
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0 votes
0 answers
74 views

Numerical integral in complex polar coordinates

point = {m -> -3, \[Nu] -> -0.2} ...
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1 vote
0 answers
54 views

How to prove the inequality of undefined function?

For example, how to prove the following inequality? ...
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  • 1,365
1 vote
1 answer
95 views

Non- linear ODE

I was working on this equation but I can't get out something for this: ...
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4 votes
3 answers
549 views

Infinite sum bug

Why: Limit[Sum[Sqrt[-i^2 + n^2]/n^2, {i, 1, n}], n -> Infinity] (* 0 *) but: Integrate[Sqrt[1 - x^2], {x, 0, 1}] (* Pi/4 *) ...
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  • 1,365
0 votes
1 answer
51 views

Adding Assumptions in double integral written in "math input" mode

I've the following double integral and I'd like to write Assumptions->Re[c]>0, but if I try to do it, I get a syntax error. How can I add the above condition but without changing the ...
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0 votes
1 answer
90 views

The prime superscript didn't work as derivative

I am really new to Mathematica. I tried a lot of ways but I failed I am using the Adomian polynomial to decompose a function. The polynomial is generated by: Then in my case I need to decompose ...
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  • 3
0 votes
3 answers
111 views

How to convert Integrate[Integrate[...] ...] into multiple integral

The following two expressions are identical from a human point of view: a = Integrate[f[x, y], x, y] b = Integrate[Integrate[f[x, y], x], y] The same goes here: <...
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3 votes
1 answer
149 views

Manual Runge-Kutta 2 for a system of 4 ODE's

A follow-up on this question : I have the following system: $\frac{dx}{dt} = p_x \\ \frac{dy}{dt} = p_y \\ \frac{dp_x}{dt}=-\frac{\partial V}{\partial x} \\ \frac{dp_y}{dt}=-\frac{\partial V}{\partial ...
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  • 199
1 vote
1 answer
134 views

Explicit integration of PDE running very slowly

I would like to solve the PDE $\frac{\partial u}{\partial t}=(1-x)\frac{\partial u}{\partial x}$, in the interval $[0,1]$, with initial condition $u(0,x)=e^{-218(\frac{-2}{3}-ln(1-x))^2}$, by manually ...
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  • 199
0 votes
1 answer
86 views

ODE solution for a simple case

I was trying to solve this but I can't not fix the problem :( please help me. a,b,f and d positive vale. ...
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2 votes
1 answer
119 views

Writing a system of ODEs in vector form

I have the following system: $\frac{dx}{dt} = p_x \\ \frac{dy}{dt} = p_y \\ \frac{dp_x}{dt}=-\frac{\partial V}{\partial x} \\ \frac{dp_y}{dt}=-\frac{\partial V}{\partial y}$ ,where $V(x,y) = \frac{1}{...
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