All Questions
Tagged with calculus-and-analysis functions
270 questions
34
votes
4
answers
26k
views
Finding Limits in several variables
Is there a way to find a limit of a multivariable function, like
$$\lim_{(x,y)\to (0,0)} f(x,y)$$
with Mathematica?
When $f$ is continuous, we can use
$$\lim_{(x,y)\to (0,0)} f(x,y)=\lim_{(x,0)\...
34
votes
5
answers
16k
views
How can I differentiate numerically?
Mathematica has two ways to integrate: Integrate and NIntegrate.
But what about D? ...
- 1,759
21
votes
5
answers
15k
views
How to find the domain and range of a function with Mathematica?
I'm studying calculus and in some exercises I am asked to find the domain and range of a function. Does Mathematica have already a built-in function for this?
I can imagine some ways of doing so, ...
- 5,573
18
votes
3
answers
4k
views
Is there a built-in function which detects singularities in a function?
Given a function f[x] and a region M in the complex x-plane, how can I find singularities of f in this region, i.e., issue a ...
- 13.1k
16
votes
2
answers
8k
views
Derivative of the Absolute Value Function in Mathematica looks very different from the actual answer
Derivative of the Abs Function in Mathematica looks very different than the actual answer. Why? How to correct this?
For example I entered ...
- 261
15
votes
1
answer
318
views
Why do big-O terms disappear in definite integrals since Mathematica 9?
In Mathematica 8, when I computed the following input:
Integrate[Series[Cos[x], {x, 0, 2}], {x, 0, a}]
Mathematica returned an expression that had a O[a^4] in ...
- 753
14
votes
1
answer
473
views
How do I get a general form for n-th derivative of a function? [duplicate]
I tried the following but it basically returns unevaluated:
...
- 243
13
votes
9
answers
6k
views
Finding maximum or minimum of implicit functions
is there any built in function that can be used to find maximum or minimum of implicit functions?
For example, if we have the equation
$$x^2 + y^2 = (2 x^2 + 2 y^2 - x)^2,$$
then we can visualize the ...
- 20k
13
votes
1
answer
664
views
solve ODE with divergencies
The solutions of the second order differential equation
$$\frac{1}{\eta}\frac{d}{d\eta}\left(\eta \frac{df}{d\eta}\right)+\left(1-\frac{s^2}{\eta^2}\right)f-f^3=0$$
is shown in Fig. 5.2 below, for ...
- 901
11
votes
3
answers
8k
views
How to find the non-differentiable point(s) of a given continuous function?
For example, the non-differentiable point of the function $f(x)=|x|$ is at $x=0$.
How to find the non-differentiable points of a continuous function that is defined numerically?
- 1,560
10
votes
2
answers
3k
views
Derivative of conjugate multivariate function
I have a problem with Mathematica, taking the derivative of the conjugate of some function. I know that a similar question has been posed before here, but the solution did not work for multivariate ...
- 103
10
votes
3
answers
682
views
Make Sinc'[0] return 0 instead of Indeterminate
I want Sinc'[0] to return 0, but instead it returns Indeterminate.
I've tried
Unprotect[Sinc]
Unprotect[Derivative]
Derivative[1][Sinc][0] ^= 0
But it doesn't ...
- 1,398
10
votes
1
answer
736
views
Is it possible to find a limit of a sequence given by its recurrence relation?
I need to calculate a limit of a sequence given by its recurrence relation. I tried the following:
...
- 437
10
votes
1
answer
714
views
Slot number cannot be filled
I got a strange message with the derivative of a two-argument function defined by a definite integral. For example if "f" is defined as:
...
- 101
9
votes
3
answers
1k
views
Inverting a function in a certain region
InverseFunction works well for globally invertible functions, like
f = 2*# + 2 &;
InverseFunction[f]
...
- 15k
8
votes
5
answers
9k
views
Find function inverse
I'm trying to find the inverse of a function:
(30*x^2 (1 - x)^2) (* where 0<x<1 *)
I tried all the following options:
1.
...
- 193
8
votes
1
answer
517
views
ReplaceAll[] and Limit[] don't give correct results for this expression under extreme variables [duplicate]
Possible Duplicate:
Funny behaviour when plotting a polynomial of high degree and large coefficients
1/x^2 + (3 + x)/(6 (1 - Exp[x] + x))
——This is a ...
- 68.4k
8
votes
1
answer
8k
views
Using Integrate to define a function
I'd like to define a function by way of the output of a definite integral with symbolic bounds. For instance, F[m_,k_] := Integrate[x,{x,m,k}] would define ...
7
votes
4
answers
3k
views
7
votes
1
answer
258
views
Checking what a function depends on
I'd like to define specific partial derivative operators that change based on if the passed expression is a function of, for example, x vs. ...
- 73
7
votes
1
answer
308
views
How to use Slater Type Orbitals as a basis functions in matrix method correctly?
This question is a continuation of my previous series of questions about basis functions.
I would like to find the minimum energy of Coulomb potential motion using matrix method.
$H=-\frac{1}{2}\Delta-...
- 1,893
7
votes
1
answer
256
views
Keeping Phase Factors in Sqrt
I am trying to plot certain holomorphic functions that contain square and higher roots. In the complex analysis sense, the function $f:z\mapsto z^\alpha$ for some $\alpha\in\mathbb C$ has a phase ...
- 287
7
votes
2
answers
265
views
Which version of Mathematica did DiracDelta become to the derivative of HeavisideTheta from that of UnitStep?
The mathematica's version of my coworkers is 5.0, they use UnitStep to derive formula, However, I also want to use the parallel features so used the latest version ...
- 693
6
votes
5
answers
464
views
Results in terms of a function
Let's assume I have a function $f(x)=\exp(10x^2)$, i.e. f[x_]:= Exp[10*x^2]. When I differentiate $f$ by evaluating the expression ...
- 161
6
votes
3
answers
5k
views
Curl of a second-order tensor
In Mathematica 9.0, the documentation for the Curl function states that in n-dimensions "the resulting curl is an array with depth n-k-1 of dimensions". Accordingly, if a 2-dimensional array is feeded ...
6
votes
2
answers
166
views
Efficient computation of Hopf invariant
I am trying to check using Mathematica the following result:
$$-\frac{1}{4\pi^2}\int_{0}^{2\pi}\int_{0}^{2\pi}\int_{0}^{2\pi} \mathbf{F}\cdot\mathbf{A}\;dk_x dk_y dk_z=1,\quad\text{where}$$
$$z_{\...
- 163
6
votes
1
answer
113
views
Indefinite order of differential operator
In the latest version of Mathematica, D can compute derivatives of symbolic order, that is, the $n^{\text{th}}$ (partial) derivative, returning an expression ...
- 9,883
6
votes
1
answer
4k
views
How to take the derivative w.r.t. an arbitrary function?
EDIT: The answer posted by @Jens works for built-in functions only. However I'm starting to wonder if this is related to my Mathematica install somehow. I have version 7. Here is an example in the ...
- 115
6
votes
0
answers
267
views
Strange Integrate behavior (a bug!)
The following two calculations should give the same result. After all, integration is a linear operation.
I have pasted the code below in case you want to play with it.
...
- 6,458
5
votes
6
answers
344
views
How to use other methods to realize `Differences`
I know the following method can get the desired result {b-a,c-b,d-c,e-d}:
Differences[{a, b, c, d, e}]
But I want to know more ...
5
votes
3
answers
584
views
Finding Saddle Points [closed]
How many saddle points are on the graph of $h(x,y)=x^3-12xy+8y^3?$
I need to find the critical point of this function, but I get stuck when I have to find the partial derivatives of fx and fy.
How can ...
- 51
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)?
...
- 3,464
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,...
- 139
5
votes
0
answers
134
views
Create custom definition for chain rule of partial derivative of a vector-valued function
I'm trying to calculate derivatives of expressions involving an unknown function $F: \mathbb{R}\times\mathbb{R}^n\times\mathbb{R}^n \rightarrow \mathbb{R}^n$, with respect to a parameter $h \in \...
- 191
4
votes
2
answers
2k
views
How to obtain numerical solutions of a two-variable equation?
I have a two-variable function:
f[x_, y_] := (4 y^2 - 6) (Cos[y π] Cos[x] - Cos[2 y π] Sin[x])
How can I find all the numerical solutions for $x$ and $y$ such ...
user67849
4
votes
3
answers
302
views
How to find absolute minimum with NMinimize
So I have a trefoil and twisted cubic that are very close to intersecting but do not. I know NMinimize can be used to best approximate the minimum, but it ...
- 91
4
votes
3
answers
307
views
Plotting a parametric function on xy, xz, and yz planes
I'm trying to plot a trefoil with a parametric function in Mathematica. How would I do that? I know I can use Parametric3D to depict all of it but how do I project it on the XY, XZ, and YZ planes?
It'...
- 91
4
votes
4
answers
414
views
Using brackets with limits in evaluation of definite integral
How to do this in Mathematica:
$$\int_1^2 x^2 \, dx=\left[\frac{x^3}{3}\right]_1^2$$
or
$$\int_1^2 x^2 \, dx=\left.\frac{x^3}{3}\right|_1^2$$
Also is there a function that represents the right hand ...
- 25.1k
4
votes
2
answers
691
views
Mathematica gives an unexpected answer for Integrate [closed]
I need to integrate the following:
\begin{equation}\tag{1}
\frac{\sqrt{C + (1 - C) x^3}}{x},
\end{equation}
where $0 < C < 1$ and $x$ is a positive variable (then $x^3 \ge 0$). When I integrate:...
- 4,133
4
votes
1
answer
407
views
Real and imaginary parts of complex logarithm
I need to obtain the real and imaginary parts of the below expression with Mathematica:
$(- i a - m^2) \ln(\frac{i m^2}{2 a})$,
where $a$ and $m$ are real.
So we have:
...
user67794
4
votes
1
answer
358
views
Volume within parameter space
Imagine a parameter space with variable 0<p<1, 0<e1<1/2 and 0<e2<1/2.
...
- 1,605
4
votes
1
answer
504
views
Jacobian Matrix?
I am doing
r[psi_, th_, phi_] := {Cos[(phi + psi)/2]*Cos[th/2],Cos[(phi - psi)/2]*Sin[th/2], Sin[(phi - psi)/2]*Sin[th/2], Sin[(phi + psi)/2]*Cos[th/2]}
and ...
- 263
4
votes
2
answers
270
views
How to calculate volume under $ f(x, y) $?
I have the following function:
σ = 0.3;
μ = 5;
cy =0;
cx=0;
f[x_,y_]=Exp[-(1/2)((Sqrt[(x-cx)^2+(y-cy)^2]-μ)/σ)^2]/(σ*Sqrt[2*π])
Plotting this with:
...
- 291
4
votes
1
answer
374
views
Inverse of vector-valued function
Having a differentiable function $f : \mathbb{R}^2 \longrightarrow \mathbb{R}^2$ such that $\det D(f)$ (Jacobi determinant) does not vanish.
How can we get the inverse function $f^{-1} : \mathbb{R}^2 \...
- 1,005
4
votes
2
answers
7k
views
How does one define a function that takes vectors as arguments but also derivatives?
Summary:
tldr; I just want to implement a version of the following mathematics function:
$$ f(a) = c_1 e^{-(a - t_1)^2} + c_2 e^{-(a - t_2)^2} + c_3 e^{-(a - t_3)^2}$$
in mathematica, and be able ...
- 275
4
votes
1
answer
151
views
How to apply RootLocusPlot correctly
I have to analyze the complex regions of variable c which is define obtained with InverseLaplace help in the form
...
- 1,129
4
votes
1
answer
4k
views
Integral of Lorentzian yields different results depending on when parameter assigments are made
I'm evaluating the integral of a Lorentzian, which I know equals one. First I define the function and evaluate the integral in two slightly different ways. Surprisingly, I do not get the right answer ...
- 1,005
4
votes
0
answers
351
views
Strange behaviour of MMA in derivatives of some standard functions
There are some peculiar things to be discovered in derivatives of some standard functions in MMA:
Strange behaviour
Example 1: Abs
We have
...
- 13.1k
3
votes
4
answers
758
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]
...
3
votes
3
answers
1k
views
Why does the first derivative of a piecewise continuous function have discontinuities?
I have this piecewise continuous function which is also continuously differentiable over time :
...
- 1,055