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
47 views

Integration not working

Can't seem to get this integration to work. s, n, b, b, l are all constants. ...
2
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2answers
69 views

How can one eliminate results in which symbols are assigned constant values?

As an exercise (pertaining to my posting Perform a constrained integration over $[-1,1]^6--yielding a "separability probability"), I issued the command ...
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2answers
68 views

Is that a bug in Integrate?

Why does the result of Integrate become negative? ...
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1answer
86 views

Write a function areaPerimeter[f,g,x] [closed]

Write a function areaPerimeter[f,g,x] which takes as input two polynomials f and g depending on the variable x, and does the following. If the curves y=f(x) and y=g(x) intersect at exactly two ...
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2answers
124 views

How to get details about how Mathematica did a definite Integral

Assuming[{Element[S, Reals],S>0},Integrate[Exp[-I*S*w]/(w^2 + 1)^(3/2) ,{w, 0, Infinity}]] gets ...
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1answer
72 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
138 views

To solve self consistent equations by simultaneously plotting them

How can we solve a pair of equations that have to be solved self-consistently (even by plotting them simultaneously) . ...
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0answers
59 views

Perform a constrained integration over $[-1,1]^6--yielding a “separability probability”

To begin with, I have a set of three positivity constraints (ensuring the positivity of a certain (twelve-parameter) $6 \times 6$ "density matrix", denoted in the standard manner as $\rho$, namely, <...
1
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0answers
124 views

Stokes theorem in Mathematica

I came across this problem in calculus. It makes a handwavy sense with the explanation provided. I have tried to prove it in MMA: ...
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0answers
65 views

Approximate/estimate the ratio of two multidimensional constrained integrals

I have a hypothesis that a certain "qubit-qutrit separability probability" should assume the value $\frac{5}{3} \left(112 \pi ^2-1105\right) \approx 0.659488$. In its initial formulation, this is a ...
2
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2answers
147 views

What is the fastest way of doing one integral in Mathematica?

I want to compute the following numerical integral in Mathematica $\int_0^L dy \int_0^y d \bar{y} f(y,\bar{y}),$ where $f(y,\bar{y})$ is a very complicated function. I show my code below where the ...
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1answer
34 views

Modify a parameter value while solving a partial derivative

I'm currently trying to do a sensitivity analysis on a variable R0 (as defined below). This sensitivity analysis aims to determine what impact has the variation of ...
2
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1answer
69 views

How do I divide an integral into parts and sum the total?

I have an exercise looking like this: Analyze the integral $4\int_{0}^{1} \sqrt{1-x^2} \, \mathrm{d} x$ numerically by deviding the interval ${]0,1[}$ into three equal parts, then summarize the ...
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2answers
71 views

Identifying total derivatives

I would like to identify which terms in my expression are a total derivative and which ones are not. For expressions in which all terms are a total derivative, one can simply use integrate to find ...
5
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2answers
374 views

Derivative of an interpolated function

I am trying to take the derivative of an interpolated function. I am given the function values and the derivatives at some irregular points. Here is my minimal working example to reproduce the error: ...
1
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1answer
114 views

Second Derivative in MMA [closed]

I would like to ask, why the following second derivative is wrong in MMA ...
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0answers
53 views

how to linearize an equation? [closed]

How does one linearize this equation?
4
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1answer
107 views

Concerning the origin of four terms in DSolve's hyperbolic solution to an ODE

Typing DSolve[{g''[y] - k^2/Gy g[y] == 0}, g[y], y] where $k$ and $G_y$ are constants gives $$\left\{\left\{g(y)\to C_1 \exp\left(\frac{k y}{\sqrt{G_y}}\right)+...
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2answers
139 views

Direct and easy calculation of the N-th derivative of a function at 0?

Does Mathematica have an easy way to calculate the n-th derivative of a function at zero? I was looking for something along the way D[e^x,{x,n,0}]. How do I do ...
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0answers
28 views

Why is an integration error appearing during this code?

I'm really confused about this to be honest. If i evaluate approx[x,k] for some k, and then input the list into the Plot function, I get a plot. But if I try to use approx[x,k] directly, I get an ...
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0answers
22 views

Acquire a single-variable function by slicing a 3d graph along a given line

Consider my Calculus 2 math problem: Given a function f[x,y]=x^2-3xy and two vectors a = {1,2} and b = {1,1}. Spanning a linear ...
2
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3answers
196 views

Compute a certain “separability probability” via a constrained 4D integration over $[-1,1]^4$

I have employed the GenericCylindricalDecomposition command (applied to $6 \times 6$ "density matrix" positivity constraints) to compute the following expression ...
0
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1answer
49 views

Replacing a derivative [closed]

If I want to calculate e.g. $\frac{dr(t)}{dt}$ then in Mathematica we can write e.g. , answer = D[r[t],t] This outputs r'[t] ...
3
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0answers
43 views

On using ConditionalExpression inside Integrate vs. getting the result, then applying Assuming on result [closed]

I thought at first that sol = Integrate[ ConditionalExpression[ integrand, conditions],....] (1) Is equivalent to and should give the same answer as ...
0
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1answer
39 views

Help with recurrence relation

I try to solve the following recursion for $n \in \mathbb{N}$. $r_i = r_{i-1} - \frac{1}{2} \cdot \sqrt{1 - \frac{4\pi^2\cdot r_{i-1}^2}{n^2} \cdot \cos^2 \left(\frac{\pi}{n}\right)}$ $r_0 = \...
1
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1answer
60 views

Midpoint Riemann Sum Code [duplicate]

I am trying to write a code in order to find the Riemann sum for the midpoint but can't seem to get the correct answer. I double checked what the answer should be but I haven't been able to find what'...
2
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3answers
94 views

Can the indefinite integral of this trigonometric function be written in closed form?

Mathematica isn't evaluating this integral and I'm wondering if there's a bug in how I'm asking: the function I want to integrate is $$f(\theta) = \cos^2(\theta)\sqrt{A \cos^2{\theta} + B \sin^2{\...
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0answers
39 views

Finding tangents and curvature of a parametric curve in 3D [duplicate]

I am in a calculus 3 class and cannot figure out how to get Mathematica to solve for unit tangent, normal tangent, binormal tangent, and curvature without getting a supper messy result. This is the ...
2
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0answers
49 views

Symbolic perturbation expansion for quantum mechanics using Hellmann-Feynman derivaties

I am interested in some quantum mechanical perturbation expansion for energies. Actually I want to implement these terms $E_n^{(k)}$. As is stated below one can do that using CAS. I would be ...
5
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1answer
103 views

Integrating function with /; in its definition

why f[x_ /; x>=0]:=x; Integrate[f[x] ,{x,0,2 Pi}] return unevaluated? Notice that the above definition of f[x] works OK ...
7
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2answers
544 views

Forcing Mathematica's Integrate to give more general answers

I have a simple gaussian integral: $\int^{\infty}_{-\infty}dx\:e^{i\alpha x^2}$. If $\alpha \in \mathbb{R}$, then: $$ \int^\infty_{-\infty} dx\; e^{i \, \alpha x^2} = \sqrt{\frac \pi {-i \alpha}} \...
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1answer
99 views

ImplicitRegion vs DiracDelta [closed]

I've been having troubles with numerical integration over implicit regions, so I checked in a simple example if the result coincides with integrating with a DiracDelta function, and found this rather ...
2
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0answers
96 views

Can this integral equation problem $\int_\Gamma \frac{e^{ik|x-y|}}{4\pi|x-y|}\varphi(y) \, \mathrm{d} y = u_{x_0}^{in}(x)$ be solved?

I am not sure if Mathematica is capable of solving integral equations in 2D/3D. I found this page in the documentation, but this is just for 1D. The following is what I would like to solve, it can ...
2
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0answers
48 views

Make Mathematica evaluate an integral symbolically

The following integral has a symbolic solution, which MMA does not return. $\kappa(x) \begin{cases} \text{mod}(x,1)\le0.5: & 1\\ \text{mod}(x,1)>0.5: & 0 \end{cases}$ $f(z)=\int_0^1 \...
1
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2answers
132 views

Multivariate series for approximating implicit system

I'm trying to approximate the solution of an implicit set of equations by means of a Taylor series. I have managed to do so for a solution expressed in terms of a single independent variable, by using ...
0
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0answers
28 views

Linear dependency of vectors of functions

I would like to find a symbolic way to obtain a Q-function (Boolean output) for the following problem. I have the following inputs: ...
0
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1answer
79 views

Infinite sum + integral

I have the following sum + integral to solve: $\int_0^{\infty}dt\, t^{\frac{m-3}{2}} e^{-t(a^2+b^2)} \sum_{n=-\infty}^{\infty} e^{-t(c^2 n^2 +2bcn)}$ My try: ...
4
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0answers
61 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_{...
5
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4answers
282 views

Computing surface integral on the unit sphere

I have a function $f$ which takes three unit vectors of $\mathbb R^3$ and returns a number, so $f \colon \mathbb R^3 \times \mathbb R^3 \times \mathbb R^3 \to \mathbb R$ which is defined as $f(\...
0
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0answers
49 views

How to find D[F[a(x,y,t),b(x,y,t)],t] in terms of D[F[a(x,y,t),b(x,y,t)],x] and D[F[a(x,y,t),b(x,y,t)],y]?

I have a function $F(a(x,y,t),b(x,y,t))$ whose derivative I would like to be displayed as $\frac{\partial F}{\partial t} = f(x,y,t)\frac{\partial F}{\partial x} + g(x,y,t)\frac{\partial F}{\partial y}...
9
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3answers
706 views

How to use Mathematica to do a complex integrate with poles in real axis?

I want to use Mathematica to compute the following complex integral: Integrate[Exp[I z ] 1/z, {z, -Infinity, Infinity}] Mathematica reports that it does not ...
1
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2answers
104 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_{...
1
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2answers
64 views

Verifying equality

I am trying to test equalities involving integrals. What are the possible causes for not being able to test the following: ...
0
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1answer
138 views

Definite double integral in Mathematica

I am a beginner in Mathematica. I want to perform the following definite double integral equation in Mathematica and want to compare the solution with mine. $$ A_{P-A} = R^2\int_{\theta=\frac\pi2-\...
0
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1answer
64 views

Find area under real part of complex curve without numerically integrating

I have a function defined within the domain 0<=x<=2*Pi. f = Cos[x] * (a - 1 - r*Cos[x])^1.5 on the domain: ArcCos[(a-1)/r] < x < 2*Pi-ArcCos[(a-1)/r] f = 0 elsewhere 'a' and 'r' are ...
1
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1answer
107 views

How do I draw a function along with a particular tangent line at a specific point? [duplicate]

I would like to draw a function (x/x^2+1), and at a certain point, i'd like to draw a dot along with the tangent line at that specific point. what I have tried is the following: ...
1
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0answers
55 views

Piecewise function inside Integrate - Wrong result

I tried to evaluate the following integral: ...
10
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1answer
248 views

Linearity Assumption [duplicate]

I'd like to define a set of variables of this sort: {K1[t],K2[t],...} Adding a general assumption: all such variables are linear. I'd like to do this to avoid ...
5
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1answer
250 views

Neglect higher order derivatives in expression

Say I have an expression of this sort: Derivative[5][x1][t]+Derivative[3][x2][t]+x1[t]+Derivative[2][x1][t] and so on... I would like to ask Mathematica to ...