All Questions
Tagged with derivative or calculus-and-analysis
5,242 questions
13
votes
1
answer
341
views
Wrong Limit with LaguerreL
Bug introduced in 7.0 and fixed in 10.2.0
Limit[(n! LaguerreL[n,-1]/(n^(n+1/4)/Sqrt[2]/E^(n-2 Sqrt[n]+1/2))-1) Sqrt[n], n->∞]
Mathematica (wrong) output
<...
- 3,369
13
votes
2
answers
499
views
Mathematica integration failure - new or old?
Bug introduced in 7.0.1 or earlier, persists through 10.1
Consider the following integral:
Integrate[Log[a Cos[x]^2 + b Sin[x]^2], {x, 0, 2Pi}]
This takes a ...
- 5,134
13
votes
2
answers
1k
views
How to take derivative of parameterized coordinate?
Suppose I have a vector in $\mathbb{R}^n$ but $n$ is not known in advance. I want to be able to write functions which operate on the components of that vector, and then I'd like to be able to take ...
- 295
13
votes
4
answers
721
views
Typeset unbalanced brackets
In LaTeX, you can typeset expressions containing unbalanced brackets, by balancing them with an invisible delimiter \left. or ...
- 2,432
13
votes
1
answer
390
views
Series with ArcTan gives wrong symbolic answer in Wolfram Language
Bug introduced after 9 and persisting through 13.1. Resolved in 13.2
Recently, I have found a very bad problem with Wolfram Language. It gives the wrong answer for a quite simple expression!
When ...
- 231
13
votes
1
answer
384
views
Extra factors appear when evaluating Euler integrals
Note: this is fixed in version 9.
When I perform the double integral in Mathematica,
Integrate[(x (1 - x))^z (y (1 - y))^z, {x, 0, 1}, {y, 0, 1}]
which should ...
- 161
13
votes
1
answer
314
views
Why do singularities disappear when performing convolutions with the Dirac Delta function?
Bug introduced in 9.0 or earlier and fixed in 10.4
I am attempting to perform a convolution involving the Dirac Delta function:
$\int_{-\infty}^{\infty} \frac{1}{t+1} \cdot \delta(t+1)\ dt$
I would ...
- 181
13
votes
1
answer
565
views
Why does Mathematica give an incorrect answer to this multiple integral?
This is not a new problem but I would like to understand why Mathematica gives the result that it does. (Volume of a hypersphere)
...
- 207
12
votes
6
answers
12k
views
Notation of partial derivative
I want to write partial derivatives of functions with many arguments. Why is it that when I type
f[x,y] ctrl+6 (0,1)
it turns out to be bad syntax? The output of
...
- 7,075
12
votes
7
answers
4k
views
Trying to prove that $x\sin(\frac{\pi}{x})\ge\pi \cos(\frac{\pi}{x})$ for $x\ge 1$
As an intermediate step, consider the function:
f[x_] := x Sin[Pi/x]
I want to prove that this function is increasing for $x\ge 1$. This can be done with the ...
- 737
12
votes
2
answers
9k
views
Equation of a line that is tangent to a curve at point
A common problem in the derivative section of calculus texts is "find the equation of the line that is tangent to the curve $y = \ldots$ at the point $P$."
To find the line that is tangent to $y = 2 ...
- 959
12
votes
6
answers
2k
views
Integrate gives wrong results
Integrate[a/(Sin[t]^2 + a^2), {t, 0, 2 Pi}]
$$\int_0^{2 \pi } \frac{a}{a^2+\sin ^2(t)} \, dt$$
gives $0$
This cannot be true. What is going on?
If I insert a ...
- 1,273
12
votes
2
answers
521
views
Total flux of the gradient of the numerical solution of a PDE through a surface
Trying to solve the following PDE with BC T==1 on a spherical cap of a unit sphere and T==0 at infinity (approximated as ...
- 597
12
votes
2
answers
11k
views
How do I find line integrals?
For example, how can I calculate
$$\int_{\left | z \right |=1}\frac{dz}{z}$$
I know that the answer is $2\pi i$ but how do I do it using Mathematica?
- 143
12
votes
3
answers
8k
views
Laurent series expansion
Can someone share how to find the Laurent series expansion of
$$f(z)=\frac{1}{(z^2-1)(z^2-4)}$$
centered at $0$ on the annulus $1<|z|<2$?
- 15k
12
votes
4
answers
1k
views
Converges or diverges?
Studying the behavior of the series $$ \sum _{n=1}^{\infty } \left(1-\frac{\log (n)}{n}\right)^{2 n}$$,
I try in 12.3.1 on Windows 10
...
- 29.1k
12
votes
2
answers
673
views
Can we abuse notation and write equations in differential one-form?
Consider a simple equation with a one-form on both sides (mathworld.wolfram is also aware of this):
$$ y\mathrm{d}x = \mathrm{d}y $$
This is a perfectly valid abuse of notation. We can carry the $y$ ...
- 11.5k
12
votes
2
answers
658
views
Cubic equation and tangent circle
I have a cubic equation as below, which I am plotting:
Plot[(x + 1) (x - 1) (x - 2), {x, -2, 3}]
I like Mathematica to help me locate the position/equation of a ...
- 1,491
12
votes
1
answer
1k
views
1/x integration giving 0 instead of infinity [duplicate]
I stumbled across the fact this integral:
Integrate[1/y, {y, 0, 1}, GenerateConditions -> False]
returns
0
Anybody ...
- 121
12
votes
3
answers
778
views
Elliptic Integrals: Mathematica and Gradshteyn and Ryzhik
In Gradshteyn and Ryzhik, (specifically starting with the section 3.13) there are several results involving integrals of polynomials inside square root. These are given in terms of combinations of ...
- 420
12
votes
1
answer
13k
views
How to Take Derivative of Interpolating Function? [closed]
So I have two numerical differentialequations which yield a InterpolatingFunction each, potentialup and potentialdown. I have ...
- 233
12
votes
1
answer
663
views
Stop the Zeta function from evaluating
I have a function F whose power series i want to find using Series.
It gives me the result I want, except there is a term of $\...
12
votes
1
answer
2k
views
Laplacian and DiracDelta
It is known that:
$\nabla^2 \dfrac{1}{|r-r'|} = - 4 \pi \delta^3(\vec{r}-\vec{r}')$.
If I do that with Mathematica, I find:
...
- 6,837
12
votes
3
answers
639
views
Assumptions allowing to calculate an elliptic integral
When I feed Mathematica the following integral:
Integrate[Sqrt[(A - x) (B - x)/x], {x, 0, B}]
it spits it back out without evaluating it. However, it can ...
- 121
12
votes
2
answers
4k
views
Maximizing a function with assumptions
Using
f[s_] := Log[(s/r)^α ((α - 2) n0 r^α + 2 π Pmax ρ r^2)
/((α - 2) n0 s^α + 2 π Pmax ρ s^2)]/s
When I run the following line:
...
- 4,069
12
votes
2
answers
928
views
Why does Arg'[1. + I] return -0.5?
From the document we know that
Arg[z] gives the gives the argument of the complex number z.
Then how about ...
- 68.4k
12
votes
3
answers
5k
views
Algorithm for parts integration
Sorry if this is a duplicate, I've searched how to do this to no avail.
What I'd like to do is a function that integrates by parts $n$ times, i.e
$$
\int u(x) v(x) dx = u \left(\textstyle{\int}v\...
- 1,619
12
votes
1
answer
2k
views
What Does "Conditional Expression" Mean When it Appears in the Output?
My grandson and I were playing around with some calculus, using Mathematica to find maxima, minima, & points of inflection for various algebraic functions (e.g., $y = 2 x^3 + 3 x^2 + 5 x + 2$, etc....
- 551
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, ...
- 245k
12
votes
1
answer
394
views
Getting wrong limit with Bessel
Bug introduced in 7.0 or earlier and fixed in 11.3
I computed a following limit (related to the asymptotic expansion of the sequence A000009 - number of partitions ...
- 3,369
12
votes
1
answer
1k
views
How to do a line integral visually in version 10 or later
In version 10, Mathematica introduces some Region functions, which allow us to do integrals very visually. We can do this even if we don't know any Mathematica syntax and it can help me show others ...
- 27.2k
12
votes
1
answer
440
views
Incorrect evaluation of integral involving a DiracDelta, whose argument has infinitely many zeros
Bug introduced in 9.0 or earlier and persisting through 12.0; reported as CASE:2323185
Let's say $X>0$ is a random variable with probability density $p_X(x)={\rm e}^{-x}$. Define the random ...
- 223
12
votes
2
answers
694
views
Why Mathematica cannot apply The Fundamental Theorem of Calculus automatically?
For some reason Mathematica cannot evaluate this definite integral:
...
- 7,333
12
votes
2
answers
498
views
Why do I get a wrong result from SeriesCoefficient?
Bug introduced in 7.0.1 or earlier and fixed in 13.3
Consider the following code:
func[x_] = Sin[x^3]/(x - 1/3);
c[n_] = SeriesCoefficient[func[x], {x, 0, n}]
<...
- 7,182
12
votes
1
answer
211
views
Something wrong when I apply Derivative on a SparseArray
Bug introduced in v5.0 or earlier, and persisting through v11.0.1 or later
There, I create a SparseArray using command
...
- 183
12
votes
1
answer
2k
views
How to do matrix calculus in Mathematica?
I would like to differentiate Ax with respect to A where A is a matrix. Now, obviously, I know the derivative is x but for more nasty expressions, is there a way to differentiate expressions with ...
- 375
12
votes
1
answer
540
views
Dirichlet coefficients as limits: wrong
Perhaps I should have included the word "bug" in my question. Here we go
There is a bug in this Limit
Limit[3^s (-1 - 2^-s + Zeta[s]), s -> ∞]
(* 0 *)
which ...
- 13.1k
12
votes
1
answer
1k
views
Contour Integration along a contour containing two branch points
I need to compute following contour integrations:
$$f(u)=\oint_\alpha dz \sqrt{z^3+z+u} \qquad ; \qquad g(u)=\oint_\beta dz \sqrt{z^3+z+u}$$
In which $\alpha$ and $\beta$ are two contours in ...
- 221
12
votes
5
answers
610
views
Limit of sequence of functions behaving strange
I'm trying to determine the limit of the sequence of functions
$$f_n(x)=\left(\frac{1}{\pi}\arctan(n x) + 1/2\right)^n.
$$
I define
...
- 121
12
votes
1
answer
1k
views
Integral with HeavisideTheta takes too long to evaluate
Bug introduced in 9.0 or earlier, persisting through 13.2 or later
I tried to compute $$\int_{-1}^1 d x_1 \int_{-1}^1 d x_2 \int_{-1}^1 d y_1 \int_{-1}^1 d y_2 \theta(x_1 x_2 + y_1 y_2)\,$$ where $\...
- 283
12
votes
0
answers
261
views
Reduce: Var is not a valid variable
Bug introduced in 7.0 and persisting through 13.1 or later. Fixed in 13.2.0 or earlier.
I am trying to integrate the following but it returns a Reduce::ivar ...
- 1,907
11
votes
3
answers
7k
views
Using D to find a symbolic derivative
I need to do the following:
Define a function
Take the derivative of this function and have a look at the symoblic representation
Substitute in some values
With the bonus that I want to use the ...
- 71.2k
11
votes
6
answers
2k
views
Can Mathematica tell me if a polynomial has all real roots?
I am trying on this polynomial,
...
- 1,191
11
votes
3
answers
697
views
Symbolic integration fails while numerical integration succeeds
I am hoping to evaluate the following integral
Integrate[((r^3 - 7)^(2/3)*(1 - (r^3 - 7)^(2/3)/r^2))/r^3, {r, 2, Infinity}]
but Mathematica informs me that this ...
- 315
11
votes
3
answers
7k
views
How do I evaluate a double or triple integral over a region?
Say I need to evaluate the integral $\iiint_W f(x,y,z) dx dy dz$ and $W$ is a region given to me like $W = \{ (x,y,z) : 1 \leq x^2 + y^2 \leq 4, 1 \leq z \leq 5\}$. I don't how to do this with a ...
- 221
11
votes
6
answers
840
views
How to calculate this integral with highly oscillating integrand?
I mean
$$\int_ 0^{2\pi}\frac {\sin (10050 t)\sin (10251 t)\cos (2022 t)}
{\sin (50 t)\sin (51 t)} dt.$$
Here is my unsuccessful trial:
...
- 29.1k
11
votes
2
answers
1k
views
Series expansion of an inverse
I have to find the series expansion of the inverse function of : $\arctan\left(\frac{\ln(1+x)}{1+x}\right)$
How do I find out the series expansion of any inverse ?
Note: The inverse of a function $f$...
- 2,549
11
votes
3
answers
421
views
Integrating a BesselJ integrand to obtain the same result as Maple 16
I would like to check the following integration:
Integrate[y Integrate[1/x BesselJ[1, x Exp[I π/4]] BesselJ[1, x Exp[-I π/4]],
{x, 0, y}], {y, 0, r}]
...
- 145
11
votes
7
answers
5k
views
Is there a way to rewrite integrals in Mathematica using u substitution?
In Mathematica, can I give it an integral and a few substitution rules and have it rewrite the integral in terms of those variables?
user176201
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