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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0
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0answers
41 views

Find inverse of a complicated function

I am looking for an analytic inverse function for ...
2
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2answers
99 views

Unexpected limit result (was: How to fix the limit problem?)

MATHEMATICA fails in proving that Limit[ Integrate[ x^(x - 1) , {x, t, 2*t} ] , t -> 0 , Direction -> "FromAbove" ] == Log[2] How to fix ...
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1answer
38 views

Plotting cylinders and planes on the same graph

How would I plot the cylinder $y^2 + z^2 = 9$ and the planes $x = 0$, $y = 3x$, and $z = 0$ in the same graph and in the first octant. ...
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1answer
35 views

How to graph a solid

Given a solid described by $S = \{(x,y,z)|0\le x\le4-y^2, -2\le y\le2, 0\le z\le y+2\}$, how do I graph the solid? I have already tried a RegionPlot3D but I might ...
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0answers
65 views

Computing differentials on a manifold

Consider $\phi:SO(3) \to \mathbb{R}^3$, $R \mapsto (R^\top e_3)\times e_3$ where $R$ is a real $3\times 3$ orthogonal matrix and $e_3 = [0\ 0\ 1]^\top$. Can Mathematica compute the differential of $\...
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36 views

Index of cumulative distribution function

When trying to compute this ...
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0answers
95 views

How to integrate this expression one time? [duplicate]

How can I integrate this expression? I want to have the second derivative of V i.e. V''[r]. Note that, you can multiply it be ...
2
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1answer
104 views

Evaluating an integral symbolically

I am trying to integrate following integral symbolically via integrate command: $$(0.09)\Bigg[1 + \Bigg\{ \int_{3t-4k-7}^{2k+7/3} (0.09) \text{exp}\Bigg(\int_{3s-...
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0answers
56 views

Finding area between curves formed by csv data

I have a bunch of Differential Pulse Voltammetry data I am trying to import into Mathematica, and find the area between each curve and the blank. So far I have used the following commands: ...
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2answers
88 views

How to correctly inter-convert between two related infinitesimals?

The following input text Clear[ρ, ν, λ, T, h, c, kB] λ[ν_] := c/ν ρ[ν_, T_] := (8 πh/c^3) (ν^3/(Exp[h ν/(kB T)] - 1)) \[DifferentialD]ν ρ[ν, T] ρ[λ, T] is ...
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0answers
16 views

Unknown Vector Valued Function

I am trying to evaluate this: D[f[x[t, k, l], t, k, l], k, l] where x[t,k,l] is a 2x1 vector and the output of f is 2x1 as well. Currently the function just outputs it as if it were a scalar ...
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1answer
91 views

Why are some of my assumptions ignored?

In my opinion Mathematica should be able to further calculate and simplify the output of G1[...]. ...
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0answers
162 views

Workaround for DSolve in V 12 when it gives undefined as solution to 1D heat PDE?

Reported to WRI CASE: 4278450 Comparing the following, all done from clean kernel The strange thing is that V 12 can solve this same PDE without the assumptions ...
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0answers
58 views

Integral of the derivative does not work

In 12.0.0 for Microsoft Windows (64-bit) (April 6, 2019) writing: Integrate[D[ArcTan[Sqrt[1 + x^5]], x], x] I get: ArcTan[...
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0answers
45 views

Integration evaluation running forever

I am trying to integrate a function but the evaluation goes on and on forever. Here is the code: ...
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1answer
38 views

Integrate ignores Assumptions [closed]

I'm asking Mathematica to compute: ...
3
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0answers
109 views

How to test whether an improper integral converges or diverges?

To verify the improper integral $\int_{1}^{\infty}x\cdot\left | \sin (x^4)\cdot\sin x \right |dx$ converges or diverges,I code ...
3
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2answers
69 views

Plotting the derivative of NDSolve divided by the solution

I'm trying to solve the Friedmann equation as is given by: $$ H^{2}=\Big(\frac{\dot{a}}{a}\Big)^{2}=H_{0}^{2}\big(\Omega_{m,0}a^{-3}+\Omega_{r,0}a^{-4}+\Omega_{\Lambda,0}a^{2.3}+\Omega_{k}a^{-2}\big),...
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0answers
80 views

Hessian of a loss function for a neural netwrok

Is a way to calculate the Hessian of the loss function in Mathematica. It's easy to define a loss function for neural nets but I couldn't figure out how to calculate the Hessian(more precisely I ...
2
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1answer
30 views

How to manipulate a plot of partial derivatives of self-defined bivariate functions?

I have a piecewise linear function (spline) describing a progressive income tax schedule, with higher marginal rates kicking in for higher and higher incomes. I want to understand the implications of ...
1
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1answer
51 views

Evaluate a nested Gaussian integral

I would like to evaluate the following integral. I suspect it just doesn't have an analytic form but thought I'd check to see if there's anything I'm missing, $\int_{-\infty}^{\infty} \mathcal{N}(z;\...
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1answer
49 views

How to solve this equation by Solve?

I have an equation to be solved. But Mathematica does not work for it. I hope the solution x can be expressed as a function of a and b ...
1
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1answer
37 views

how can I plot two variables (inside a definite integral) against each other

I want to plot x against y in this relation:
2
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1answer
28 views

How to mark non-pre-defined functions as functions of a certain variable during calculation of derivative?

If I have some calculation $X=v\cdot\delta$ and I want to calculate $\frac{\partial}{\partial v}X$, the result according to Mathematica is $\delta$. Now, I want to define $\delta$ as a function of $v$ ...
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0answers
32 views

Why do I get the error message as integral does not converge?

Can anyone please help me to find the integral value of the following expression, Integrate[(x - t)^(-1), {t, 0, x}] It will be of a big help to me.
3
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1answer
72 views

Evaluating an integral combining a Bessel function with some other functions [closed]

How can I evaluate the integral of $j_1 ^2(x)\exp(-bx)/x$ from 0 to ∞?
5
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1answer
73 views

Variable dimensional integrals

I need to Integrate a function, but let the number of dimensions vary. Integrate[Product[Exp[-( t[[i]] )^2], {i,1,4}], options] ...
2
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2answers
103 views

Evaluating FourierTransform like integral manually

I am trying to evaluate Integrate[x^2*Exp[I k (x - 1)], {k, -∞, ∞}, {x, -∞, ∞}] Since $\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{ik(x-1)} d{k}$ is $\delta(x-1)$, ...
1
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1answer
36 views

Limit of an integral and hypergeometric function

I want to evaluate the following integral: Integrate[Sin[θ]^(D1 - Nc - 1)/(A Cos[θ] - I ϵ)^(N1 - Nc), {θ, 0, π}, Assumptions -> A > 0 && ϵ > 0] ...
2
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2answers
138 views

Compute unstable integral with high precision

I want to calculate $$ \int_{\mathbb{R}^n} e^{-t_1^4-t_2^4- ...-t_n^4-(1-t_1-t_2-...-t_n)^4} \text{d}t_1 \text{d}t_2 ... \text{d}t_n $$ with the highest possible n. I tried with ...
3
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4answers
442 views

How can I simplify $\lim_{n\to \infty } \, \int_0^n e^{-x} \left(1-\frac{x}{n}\right)^n \, dx$?

How can I simplify $\lim_{n\to \infty } \, \int_0^n e^{-x} \left(1-\frac{x}{n}\right)^n \, dx$ ? Limit[Integrate[(1 - x/n)^n/E^x, {x, 0, n}], n -> Infinity] ...
5
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3answers
492 views

How do I find and plot the intersection of these three surfaces?

I know how to find the region of intersection of these cylinders using integration, but how would I plot that in mathematica? This is what I have ...
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1answer
82 views

Problem in Integration

I want to calculate both of these integrals: Integrate[1/Sqrt[c x + Sqrt[ b x + x^2] + a], x] and ...
2
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1answer
93 views

Calculation of sum $\begin{aligned}\sum_{k = 1}^{n - 1}\end{aligned}\left(1+\cos\left(\frac{k\,\pi}{n}\right)\right)^n$

Having established that Mathematica cannot calculate the following summation: sum = Sum[(1 + Cos[k Pi/n])^n, {k, 1, n - 1}] I implemented the classic "plan B", ...
2
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1answer
88 views

When to trust Integrate?

I'm trying to get an analytical expression for the following integral: ...
3
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2answers
167 views

On Certain Characteristic Function

I am trying to find the characteristic function for Johnson's SU distribution by integrating the probability density function with Exp[I*t*x] but Mathematica is ...
0
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1answer
60 views

Is there any way to do this integral without expanding the exponential power?

Integrate[ x^(1/2) Exp[-10.7* c/x] Sqrt[(1 - (125/1000)^2) (1 - (2*4.18/125)^2) - (1 - (2 x/ 1000))^2], {x, 5.03, 994.97}] or can i do this indefinite ...
4
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3answers
108 views

Problem with minimization of $x^z$

I have plugged in the following problem Minimize[{x^z, z >= 1, z <= 10, x >= 1}, {x, z}] Can anyone explain please why the software does not provide the ...
0
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1answer
38 views

How to deal with the derivative of an variable upper limit integration by the symbolic form?

I need a rusult with the following form D[Integrate[f[t], {t, low[x], upper[x]}], x] (*Out[]=-f[low[x]] (low^\[Prime])[x]+f[upper[x]] (upper^\[Prime])[x]*) My ...
0
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1answer
36 views

Interesting output in Integrate [closed]

I found interesting behaviour, best illustrated in the following example. When trying to evaluate Integrate[1/(d^2 - 1), d] Mathematica 11.3 gives the following ...
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0answers
44 views

Fitting a Curie-Weiss law function with a diverging point to a data

I would like to fit a function to some experimental data such that the extracted value of Curie temperature (Tc) describes the temperature at which the response flattens out to a constant value. ...
0
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0answers
23 views

Problem with integrating special function under assumption

i found different results when integrating a special function (see below), depending where i place my assumption (x > 0, x0 > 0). The problem is that the two solutions are not compatible. In fact, if ...
0
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1answer
64 views

Integrate a function of NDSolve

I'm having this problem of integrating a function that's the solution of a NDSolve multiplied by another function. I basically compute a cycle wich add at each step the quantity ...
0
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1answer
55 views

Coupled PDEs: Wave and String Equations

I need to solve a system of mixed string and wave equations. Omitting some constants it looks like this: $$u_ {\text {yy}} (y, t) - u_ {\text {tt}} (y, t) = \varphi _{t}(x, y, t)$$ $$\nabla _{\{x,y\}}...
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0answers
12 views

Eingenvalues and Listplot take a long time optimize

I try to calculate the following eigenvalues of a matrix of N=5 10^3 X5 10^3 and representing it using Listplot as follow ...
1
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3answers
80 views

Summation and product term expressed in Mathematica [duplicate]

I have the following term which I would like to express correctly in Mathematica: $$\sum_{i=1}^{m}\frac{1}{\prod_{j=1,j\neq i}^m(\rho_i-\rho_j)}$$ This means that for $m=3$ for example you should ...
0
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1answer
63 views

Evaluate integral over quadratic form [duplicate]

What would be the most kosher way to evaluate an integral like this in Mathematica: $$\int_\mathbb{R^3} \exp\left((x-b)^tA(x-b)\right)dx \ ?$$ Here $A$ is a real symmetric matrix and $b$ some vector ...
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1answer
63 views

NIntegrate a singular function

I am trying to do this integral numerically: ...
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0answers
22 views

Assign a Derivate to a new fuction variable

I'm having a new problem. I'm trying to create a new function from a Derivate of another function, but i'm can't getting it. Someone could help me? I'm thinking that is something in the syntax of the ...
-1
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1answer
63 views

How to find the maximum of a function with several variables symbolically?

I want to maximize a function with respect to tau1, tau2, and alpha, and it is like the following: V + β1 τ1 - α τ1^2 - β2 τ1^2 + \τ2 - 2 τ2^2 + α τ2^2 I know I ...