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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2
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0answers
85 views

Drawing a cyclogon

Is it possible to draw the path of a point on the boundary while the polygon is rolling, or writing a function of theta similar to the cycloid parametric equations: $$ x = r (\theta - \sin(\theta)),\...
1
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0answers
33 views

How to compute Bott-Chern operator ddc

I wish I could make Mathematica compute the $dd^c=2i\partial\bar\partial$ of a function depending on several complex variables. The question involves complex or Wirtinger derivatives (https://...
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1answer
41 views

Calculation using Integrate & Plot take too long time & some error

I want to plot this equation. mu, Eb, gamma, Eg are constant parameters and x is independent parameter. ...
14
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3answers
534 views

How to evaluate $ \int_0^1{\mathrm e^{\mathrm i\pi x}x^x(1-x)^{1-x}\mathrm dx} $ and get $ \frac{\mathrm i\pi \mathrm e}{2\cdot3\cdot4} $?

The following definite integral identity holds $$ \int_0^1{\mathrm e^{\mathrm i\pi x}x^x(1-x)^{1-x}\mathrm dx} = \frac{\mathrm i\pi \mathrm e}{2\cdot3\cdot4} $$ and is easy to verify numerically. ...
0
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1answer
81 views

What is the proper way to use the Prime typesetting in an integral? [closed]

I've got an integral I'm trying to calculate in Mathematica: $$\tau(\eta)=\int_0^\eta n_e\sigma_T a\space d\eta\prime$$ My best attempt to represent that in Mathematica has been: ...
1
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1answer
42 views

Plotting a function on the different axis

I've solved the DE: $$a\cdot x(t)+b\cdot\ln\left(1+c\cdot x(t)\right)=-p\cdot x'(t)\tag1$$ Where $x(0)=k$ And I got the following solution: $$t=\int_{x(t)}^k\frac{p}{a\cdot z+b\cdot\ln\left(1+c\...
1
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1answer
63 views

visualizing branch cut of a complex function

I'm working in the plotting the branch cut of a complex function, namely: $w(z) = (2+z) \ln(2+z) - 2(1+z) \ln(1+z) + z \ln z$. To do so, I have tried this: ...
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0answers
26 views

Contracting the KroneckerDelta should bring it back

I would like to define a function $$V[\phi] = \sum_{1\leq i,j,k,l\leq dim}\lambda_{ijkl} \phi_i\phi_j\phi_k\phi_l, $$ with $\lambda_{ijkl}$ defined by contractions of the type $\delta_{ij}\delta_{kl}$...
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0answers
35 views

Wrong derivative for a function that I try to define without explicitly giving number of variables

What I actually want is to define functions depending on the number of variables which will be specified during some computation. My (probably wrong) attempt is ...
0
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1answer
58 views

Get Inverse Function in a certain domain

I'm trying to make Mathematica giving me the inverse function of $f$ (below) when restricted to the interval $[v,1]$. Using ConditionalExpression or ...
0
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1answer
112 views

Verifying that an inequality has a finite limit

I am trying to verify the following inequality using Mathematica, but without any success. $$ \lim_{a\rightarrow0^{+}}\frac{\intop_{0}^{1}e^{\frac{-30}{\sqrt[15]{a^{16}*\log\left(\frac{1}{x}\right)}}}...
4
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2answers
131 views

Why am I getting a negative value from $\int_{0}^{\infty}x^{\frac{-9}{8}}*e^{-16*x^{\frac{-1}{8}}-\frac{2*x}{0.5^{9}}}dx$?

In Mathematica 9, I want to evaluate the following integral: $\qquad \int_{0}^{\infty}x^{\frac{-9}{8}}*e^{-16*x^{\frac{-1}{8}}-\frac{2*x}{0.5^{9}}}dx$ The result should be positive. However, ...
4
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3answers
274 views

A discontinuity in an integral

I have a function which I'm trying to integrate, which is perfectly smooth and Mathematica sees it as well defined everywhere in the region over which I'm integrating it. The function is: ...
2
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1answer
194 views

Solving multi-field problems in MMA 11.3

Based on the PDE model proposed by @Schumacher Solving a second order coupled PDE system, the one dimensional multi-field Problem I would like to solve such benchmark test: namely: u(x): ...
0
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1answer
41 views

DSolve not satisfying such kind of initial conditions

I am trying to solve this equation in MMA 11.3 $2 \epsilon s^{\prime \prime}+\frac{1-s}{2 \epsilon}=0$; BCs: $s^{\prime}( \pm 1)=0$ and $s\left(0\right)=0$; The analytical solution is expressed as:$...
4
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2answers
133 views

Solve a logarithmic equation by NSolve

I'm trying to solve: $$1 - |x| Log(\frac{|1+x|}{|x|}) = 0.84$$ by using NSolve: NSolve[1 - Abs[x] Log[Abs[(1 + x)/x]] == 0.84, x] Mathematica doesn't return ...
1
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1answer
121 views

Writing a function that can perform n-times fractional Caputo derivative of order alpha to a function u(x) inductively

I wrote this code, but it fails to give the intended answer which is D^(1/2) three times of x^2. any suggestions to correct this code or write a new one.
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0answers
35 views

Define the partial derivatives of a multivariable function

I have some vector $K = [K_1, K_2,...K_n]$ which is a function of variables $(x,y)$ i.e. $K = K(x,y)$ I want to symbolically calculate the partial derivatives e.g. $\partial_x K$, $\partial_x \...
0
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0answers
107 views

Need help with symbolic integration

I have f[x_] := g[x]/g'[x] FullSimplify[Integrate[f[x], x]] I'm trying to get F[x] in terms of ...
5
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1answer
149 views

Visualizing Riemann surface (two branches) of logarithm

I'm trying to plot two branches of the complex multi-valued function: (In a previous post - linked above-, Mathematica found the branch cut of the following function between -1 and 0) $1-z\ln[(1+z)/z]...
4
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2answers
209 views

To plot branch cut of logarithm

I like to see the branch cut of the function: $$1 - z \ln[(1+z)/z].$$ If I plot it in the complex plane: ...
0
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0answers
39 views

Normalized eigenvectors are different depending on when the limit is calculated

I have this code: ...
0
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0answers
61 views

Unexpected difference between integral and summation?

I am trying to integrating something like: Integrate[Exp[-i*(k*x+k*z)]*Exp[-(x^2+z^2)],{x,-largenumber,largenumber},{z,-largenumber,largenumber}] My issue is that ...
1
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1answer
100 views

Is it possible to use mathematica to find the derivative of a function with respect to some “ax” where “a” is a constant?

Mathematica evaluates the derivative of most complicated functions within a fraction of second. But while doing analysis sometimes we might be in a situation to differentiate or integrate a function ...
0
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2answers
48 views

problem with constant variables

Variable x is supposed to be constant, but the program calculates for each variable x, 100 numbers of x. I want all my x variables to have the same definition, but the program constantly changes each ...
0
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0answers
37 views

How to obtain the gradient of a function wrt a vector?

I'm new to mathematica and I wanted to obtain the gradient of the following function: where b_k(s) is the cubic B-spline basis function associated with control point k. I tried this code: ...
9
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3answers
1k views

How can I solve for the intersection points of two ellipses?

I need to find the points at which the two ellipses implicitly defined by $\qquad y^2=4-4\,x^2 \quad[E1]$ and $\qquad (1-(x/2))^2+(y-1)^2=1 \quad[E2]$ intersect. So I isolated $y$ in E2 and ...
1
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1answer
106 views

How is the best way to integrate using loops

I'm using Mathematica, and I want to integrate a function f[wr] in wr using some method that works with a Table/Do/For in the variable wr. I tried to use something like the Riemann's sum, to evaluate ...
0
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0answers
42 views

How to solve this DE to get a matrix?

I have this DE $ g^{m \bar{n}} g^{r\bar{p}} \partial_t {g}_{m r} \partial_t {g}_{\bar{n}\bar{p}} = X_{i\bar{j}}~ \partial_t{f}^i (t) ~ \partial_t{f}^\bar{j} (t) ~~~ ~~~~~~~~~~~~(*) $ Where ...
3
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0answers
65 views

Derivative of eigenvalues

I work with 4x4 Hermitian matrices (r). I want to calculate a derivative of a function f[t,r] (ff[t_,r_]=1/2*D[f[t,r],t]), where the function f depends on the absolute value of the eigenvalues of r. ...
3
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1answer
68 views

Implementing Simpson's rule with two variables

For one of my lab project I need to integrate a given function (in this example, f1, using Simpson's ruls. When only one variable ...
0
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0answers
60 views

Discrete inverse Laplace transform

I'm trying to find a method for obtaining the inverse Laplace transform of a series of discrete data points. I have tried deriving an analytical function from the data and then using that function ...
0
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0answers
38 views

Plotting matrices

I have this expression: $\int g^{m \bar{n}} g^{r\bar{p}} \partial_t {g}_{m r} \partial_t {g}_{\bar{n}\bar{p}} ~ d^6 y = f(t)$ Where m,n,p,r=1,2,3 and the $\bar{}$ ...
0
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0answers
50 views

What is the representation of the Harmonic Number being used by Mma in this result?

The Fourier Transform of the function F[x_] = (m/Sqrt[\[Lambda]])*Tanh[(Sqrt[x^2]*m)/Sqrt[2]] where all variables are real, and $m>0$ is given by (Mma 11.0) ...
1
vote
2answers
102 views

Derivative for arbitrary number of times

I want to take the derivative of a function with respect to $ t $ for an arbitrary $ k $ number of times at the point $ t = 0 $ using Mathematica. The function is $$ f(t) = \frac{\mathrm e^{kt}\...
0
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1answer
65 views

defining rule for derivative on arbitrary function

Is there a way in Mathematica for defining rules for derivatives on undefined functions? For example, I would like to define a rule such that for any function $f(x,y)$, when I take the $x$-derivative ...
3
votes
1answer
104 views

Implementing the Peano-Baker Series with Mathematica

I would like to calculate the Peano-Baker series with mathematica, namely an integral expansion to approximate the state transition matrix: Say I have a matrix $A(t)$ and I want to calculate: $\Phi(...
3
votes
2answers
148 views

Plotting the inverse function of a complicated function

So I have a function F[x_] = Assuming[{Element[x, Reals], -1 < x < 1}, Integrate[1/Sqrt[(x^2 - 1)^2 + alpha*x], x]] I'm now interested in the ...
1
vote
1answer
37 views

Mixed partial derivatives with respect to n variables

I have the following generic set of variables: x = {x1, x2, ..., xn} nu = {0, 1, 2, ..., n - 1} order(nu) := Total @ nu; I would like to calculate: $$ \frac{\...
0
votes
0answers
170 views

Solving a quadruple integral of view factor

I'm solving an analytical quadruple integral in Mathematica, but the app can't solve it. I don't know if it is because that I do a bad formulation of the integral or maybe, Mathematica is not able to ...
3
votes
1answer
108 views

Reconstructing a function from its gradients

I have a list of the components of the gradients, $\partial f/\partial x_i$, of a function $f(x_1,x_2,\cdots)$. Is there some neat way to reconstruct the function $f$? One approach to doing this ...
5
votes
2answers
76 views

How can I help SumConvergence give the right result?

I've been trying to use the SumConvergence on the following series: SumConvergence[1/(n Log[n] Log[n Log[n]]), n] This ...
4
votes
1answer
117 views

Line-Dirichlet Boundary NDSolve

I have defined such a Dirichlet boundary conditions which fixes the nodes in a line, however, the results show that only two nodes are fixed, not the nodes in a line, Code: ...
0
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0answers
39 views

Analytically integrating bivariate Gaussian PDF

I have this somewhat complicated expression in terms of the following 8 variables (note: actually, $X$ and $Y$ are variables, and $a,b,c,d,e,f$ are longer expressions as a funtion two other variables, ...
0
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0answers
23 views

Unitary group as a region

I would like to compute the area of some subspaces of $SU(n)^d$ defined by group equation (basically $U_{i_1}^{e_1}\dots U_{i_k}^{e_k}=1$) . I know that there is the function ...
0
votes
0answers
55 views

How to use replace for only one term in a long expression in Mathematica?

I am doing some Cauchy integration and in some intermediate steps I have something that looks like that: (..many_terms...+I*\epsilon(k1-p1))/k1 where ...
0
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0answers
54 views

Mathematica gives uncessary condition for integration of $(b - s)^\alpha(s - a)^\beta$

When I compute the integral of $(b - s)^\alpha(s - a)^\beta$ on $[a,b]$ by the following command: ...
0
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0answers
35 views

High dimensional integral not evaluating? [duplicate]

I am trying to integrate as follows: ...
1
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1answer
62 views

Finding the limit of this neat solution

In an earlier post, I asked "Can Mathematica find an expression for the distribution of the median of N i.i.d. random variables?". JimB found the very neat solution: ...

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