All Questions
Tagged with derivative or calculus-and-analysis
5,242 questions
7
votes
2
answers
179
views
Can Mathematica differentiate summation notation with respect to an indexed variable?
My question is whether Mathematica can represent an expression such as:
$$y_i = \sum_{j=1}^N a_{ij} x_j$$
symbolically as a sum over indefinite bounds (unknown $N$), and differentiate it with respect ...
5
votes
1
answer
444
views
Symbolic second order derivative of real antisymmetric matrix in Mathematica
I would like to calculate the second order derivative of a function with respect to a matrix coefficient symbolically. Below is the first order derivative of S with respect to the coefficient of x. ...
2
votes
3
answers
236
views
Trouble with branch cuts plotting a real function
When I plot the following function
...
0
votes
1
answer
225
views
Symbolic partial differentiation of a Lagrangian function including a summation
I'm using Mathematica to double-check that I correctly derived first order conditions for the models I'm researching. Before I get started, I wanted to actually teach myself how to use Mathematica, ...
1
vote
2
answers
97
views
Parabolas to the right or to the left
I would like some help to make the opening (concavity) of the parabolas go to the right or left, and not up or down.
...
0
votes
0
answers
80
views
0
votes
1
answer
61
views
Computing an exact range with FunctionRange
When using FunctionRange to compute the range of the two-argument ArcTan function, Mathematica fails to return an exact answer.
...
1
vote
2
answers
136
views
Solving differential equation through function [duplicate]
Folks! I am trying to obtain the plot the phase portrait of the following differential equation in Mathematica. My task is to do the work through function by numerical solution. The equation is:
$$\...
0
votes
0
answers
38
views
Advice on how to get an approximate symbolic expression for nasty limit-integral
I have a fairly nasty integral-limit expression that I need to resolve and was hoping for some advice on the best practice to getting a solution. I have the following functions
\begin{gather*}
w(\beta,...
5
votes
4
answers
552
views
Plot an infinite but convergent series
I would like to plot the following function
$$
f(m)=-2 \sum_{k=0}^{\infty}\binom{1/2}{k} \left(2^{2(k-1)}-1\right)\zeta(2 (k-1))\left(m^2-\frac{1}{4}\right)^k
$$
I am only interested in $m$ between 0 ...
1
vote
1
answer
69
views
Parallel loop wrecks integration
I am doing a quadruple nested integral and noticed that it produces inconsistent results when in a parallel computing environment.
...
2
votes
3
answers
83
views
Strange behavior when integrating vector valued piecewise function
The following straightforward integral should clearly equal $\{0,0\}$ for any $R\in\mathbb R$:
...
0
votes
2
answers
86
views
Error on the square of sum of binary elements for symbolic upper limit
I'm seeing the following error: as a part of some equations, I have $E[(\sum_{i=1}^n X_i)^2]$, with $X_i\sim Ber(p)$. Using independence between $X_i$ and $X_j$ for $i\ne j$ and that $X_i^2=X_i$, $E[(\...
2
votes
1
answer
186
views
How to separate parameters from a list based on certain conditions?
The following codes create a table of parameters Br and Dr based on corresponding parameters b and c. How to separate/divide the list/table of Br and Dr for which corresponding b or c are negative ...
1
vote
0
answers
47
views
Seeking Upper Bound on the Number of Extrema for Large Function
I'm working on finding an upper bound on the number of extrema for the function f[h,l,d,x,y] concerning the variable h. The ...
0
votes
0
answers
64
views
How to Replace the expression with unknown parameters?
I do not want to do Derivative, because it will be very slow after million times.
My expression is just the Exponential as following,
...
0
votes
0
answers
75
views
Weird indefinite integration in Mathematica explicitly introduces a pole that does not exist
Consider the following code:
...
3
votes
2
answers
585
views
Is it possible to get analytic solution of this integral?
I am trying to evaluate the following integral, but Integrate just returns the input:
...
0
votes
0
answers
79
views
Reflected Wave and Power Spectrum for a Toda Oscillator
I'm trying to analyze the power spectrum of the reflected wave. Specifically, I want to compute the time series, take the Fourier transform, and average the absolute values to obtain the spectrum. The ...
3
votes
3
answers
226
views
Reconstruct points on a sphere from matrix of dot-products
Suppose I have a matrix of pairwise dot products corresponding to vectors on a sphere. How do I reconstruct these vectors (up to a rotation?).
Below is a solution that reconstructs a set of vectors ...
4
votes
1
answer
146
views
Why equal series written in a little bit different way both diverges and converges?
I am very new to Mathematica, so I am sorry if the answer is obvious. I wanted to calculate the sum of the following series:
...
5
votes
1
answer
226
views
Series Converges or Diverges?
I am running MMA 13.1.0.0 on Windows 10 - x86.
Is there any way to get Mathematica to conclude that this sum converges (maybe it needs some additional input)?
...
0
votes
1
answer
176
views
Concept of tangent and residual in AceFem and automation
here trying to solve the pde: ut + ux = 0 in acefem. I have two questions regarding tangent and residual.
Tangent shows the s$$ and residual shows ...
2
votes
3
answers
287
views
Help with derivative command [duplicate]
I am trying to find the Mathematica command for
$$f(k,a,b)=\frac{\partial^k}{\partial {a}^k }\csc\left(ab\pi\right)$$ and here is what typed for the command ...
2
votes
1
answer
138
views
How to find this four-dimensional integral as an iterated integral with Mathematica?
I mean the following integral $$\int\limits _{[0,1]^4}\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\,dx_1dx_2dy_1dy_2 .$$
Its value is the average length of a random interval in the unit square (see, for example, ...
1
vote
2
answers
80
views
How to extract coefficients from a rational expression [closed]
I try to extract the following coefficients from a rational expression, but something that seems so simple mathematica cannot do it $$\frac{3 \left(9 \sqrt{39}-13 i\right)}{26 \left(4 i k+3 i+\sqrt{39}...
4
votes
1
answer
177
views
Robust Computation of Inverse Stieltjes Transform
I'm interested in finding a small set of procedures which I can use to compute inverse Stieltjes Transform for a random variable. I'm interested in
Stieltjes transform
solved: ...
0
votes
2
answers
108
views
Finding the best initial condition of NDSolve for different boundary condition
I have the following Mathematica code, which solves the second-order differential equation:
...
5
votes
2
answers
295
views
Visualizing density given by its Stieltjes transformation?
How can I use Mathematica to get $f(y)$, the PDF of a random variable given its Stieltjes transform $g(s)$?
$$g(s)=E\left[1/(s-X)\right]$$
I need to visualize probability density of a random variable ...
3
votes
1
answer
191
views
Inverse Stieltjes Transform Implementation?
Are there any implementations of Inverse Stieltjes Transform in Mathematica?
For instance, in the example below, I'm hoping to recover Beta(1/2,1) density $\frac{1}{2 \sqrt{x}}$ from its Stieltjes ...
0
votes
1
answer
95
views
Unable to solve an analytic integration
I have been trying to do an analytical integration of the following function with respect to s:
...
1
vote
0
answers
116
views
Integrate numerically over implicit complex contour
I'm trying to numerically evaluate the following integral:
$$
\int_0^\infty dt\ e^{-(a+i\alpha) t}(1+e^{-t})^{-(b+i\beta)},
$$
where $a,b$ are some rational numbers and $\alpha,\beta$ are relatively ...
0
votes
2
answers
71
views
How to speed up the n-th Laplacian or derivative of function?
We wish to solve for the multiple Laplacian of a function, and here we use sum of E-exponential functions as an example:
...
0
votes
5
answers
285
views
What's the correct answer for $ \oint_{C} \frac{e^{1/z}}{z^{2} - 1} \, dz$?
$$
\oint_{C} \frac{e^{1/z}}{z^{2} - 1} \, dz,
$$
where $ C $ is the unit circle defined by $ |z| = 1 $.
The original question can be found under the answer from Ron Gordon's answer to "contour ...
1
vote
1
answer
262
views
How to obtain the n-th derivative of $1/(-x+xe^{(i \pi x)})$? [closed]
Wolfram Alpha symbolic integer derivative to this question is wrong.
How can i obtain it by Mathematica?
Thanks
D[1/(-x + x*Exp[I*Pi*x]), {x, n}]
0
votes
2
answers
66
views
How do I solve for a variable for a derivative equation?
So I'm entering this as input:
f[p_] := Subscript[l, i] Log[ p] + (N - Subscript[l, i]) Log[1 - p]
f'[p]
...
0
votes
1
answer
145
views
4
votes
3
answers
212
views
Solving integral equation numerically
I tried to solve a integral equation numerically, but I failed. The equation is:$\epsilon(\beta)=r\cosh{\beta}+\int^{\infty}_{-\infty}\varphi(\beta-\beta')\ln(1+e^{-\epsilon(\beta')})d\beta'$ were $\...
2
votes
1
answer
252
views
Why "Integrate[Abs[x1 - x2], {x1, 0, t}, {x2, 0, t}]" get wrong output?
Bug introduced in 12.0 or earlier and fixed in 14.0.
To verify the following double integral
$$
\int_0^t{\int_0^t{\left| x_1-x_2 \right|\mathrm{d}x_1\mathrm{d}x_2}}=\frac{t^3}{3}
$$
The MATHEMATICA (...
3
votes
1
answer
119
views
Why is Mathematica using Re'[..] in calculation? [duplicate]
Suppose I have the following Mathematica code:
a[x_] := 1/Abs[E^Sqrt[f[x]]]
g[x_] = D[a[x], x]
The output of g[x] is $-\frac{e^...
6
votes
4
answers
510
views
Approximation of the Fabius function with a quotient of exponentials
Approximation of the Fabius function $f(x) = \text{FabiusF}[x+1]\cdot \text{HeavisideTheta}[1-x^2]$ - FabiusF[x] doesn't work in Wolfram-Alpha
I am looking to figure out how well the displaced version ...
12
votes
3
answers
254
views
Evaluation leak in Derivative[]
Bug introduced in 4, persisting through 14.1.
I usually only report bugs, but in this case, two reason prompted me to post this:
Given that derivatives are computed internally by various commands, ...
0
votes
1
answer
92
views
A hypergeometric series function
Maple and Mathematica are very efficient to find closed form of a hypergeometric function as for example:
$$\sum _{k=0}^{\infty } \frac{(-4)^k z^k \Gamma \left(1 k+\frac{1}{6}\right) \Gamma \left(\...
6
votes
1
answer
238
views
How to force Mathematica to evaulate LerchPhi[1,0,1] to -1/2
Mathematica fails to evaluate LerchPhi[1,0,1] (it gives ComplexInfinity). Based on the relation LerchPhi[1,q,1]=Zeta[q], we ...
0
votes
1
answer
125
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 ...
5
votes
0
answers
171
views
Hypergeometric Function Integration Using Mellin-Barnes Representation
I have the following integral:
$$
I=\int_0^1 d \alpha d \beta d \gamma r(\gamma) s(\alpha, \beta) T(\alpha, \beta, \gamma)
$$
where I define
$$r(\gamma)=(\gamma(1-\gamma))^{ -1 / 2+\epsilon / 2}$$
and,...
1
vote
1
answer
96
views
How to Solve Partial Derivative with Summation (for Linear Regression)?
I'm demonstrating linear regression so I would like to calculate derivatives 1 & 2 using Mathematica, something like this
This is based on the textbook notation in Johnston's Econometric Methods ...
0
votes
1
answer
150
views
How to expand the existing basis set so that it becomes more complete?
I have a set of anisotropic gaussian basis set which describes the ground state of the system with great accuracy.
The Hamiltonian of the system has the following form: $$H=-\frac{1}{2}\Delta-\frac{1}{...
0
votes
0
answers
104
views
Speed up the integration of HeavisideTheta over the range from zero to infinity
The problem is described in the following figure. Mathematica version 14.0, OS windows 10 LTSC.
The integration interval is [0,infinity), and the value of HeavisideTheta in the point 0 is 0. we can ...
2
votes
0
answers
100
views
Why does HeavisideTheta give different value at the point 0 and 0.?
As the following figure described, why does HeavisideTheta give different value at the point 0 and 0.? Especially at the coordinate origin, one gives the value 1 and another 0 when HeavisideTheta[0]=0....