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10h
comment This one weird thing that bugs me about summation and the like
Well, Thiele expansion is the exact analog of Taylor expansion for continued fractions. In this case, instead of evaluating derivatives at your expansion point, you're evaluating reciprocal derivatives.
16h
comment This one weird thing that bugs me about summation and the like
With respect to the continued fraction: you'll want to look up Thiele expansion and reciprocal detivatives.
16h
comment Is university math all about proofs?
@John, I'll dispute that; sometimes they just have coffee instead.
May
4
comment How to integrate $\frac{1}{\sqrt{x^2+y^2+z^2}}$
Ah, right; the Jacobian is $r^2 \sin\phi$, so you cancelled everything correctly. The conclusion should now be clear.
May
4
comment How to integrate $\frac{1}{\sqrt{x^2+y^2+z^2}}$
@rekt, then your integrand times the Jacobian should only be $\sin\phi$, no?
May
4
comment How to integrate $\frac{1}{\sqrt{x^2+y^2+z^2}}$
rekt, did you remember to make the coordinate changes in your original integrand?
May
4
comment How to integrate $\frac{1}{\sqrt{x^2+y^2+z^2}}$
@Francesco, your spherical coordinate convention might just be different from the OP's.
May
3
comment Are all mathematicians human calculators?
@Mariano, Gauss did numerical analysis by hand, but of course he did not have computers, and we can't really say now if he's sane. :) On my part: I could do numerics by hand, but why should I bother?
May
3
comment A generalization of Bell numbers to arbitrary complex arguments
A related question. As I noted in the answer to the other question, one might consider using Cauchy's differentiation formula with an appropriate contour for numerical explorations. This is similar to the approach I made to generalize the partition numbers.
May
3
comment Invertible skew-symmetric matrix
To jump a bit forward: odd-order skew-symmetric matrices are necessarily singular, but even-order ones don't have to be.
May
3
comment Handling complex arguments of elliptic integrals in Maple
Most CAS suck at producing useful elliptic integral expressions. You might want to look at Byrd/Friedman and see if your particular form is in there.
May
3
comment An identity involving Bernoulli and Stirling numbers
See the first formula in page 133 of this paper; the proof of the associated theorem is in previous pages.
May
2
comment Diagonal factorization of upper triangluar matrix to unit uppper triangular matrix
Consider forming a diagonal matrix from the diagonal elements of your upper triangular matrix, and see what happens if you premultiply the inverse of this diagonal matrix to your triangular matrix.
May
2
comment How to put 9 pigs into 4 pens so that there are an odd number of pigs in each pen?
Made my day. Thanks for the giggles!
May
2
comment Calculation of Integral of $ \int \sqrt{\sec 2x-1}\;dx$ and $ \int \sqrt{\sec 2x+1}\;dx$
Because sometimes, the CAS returns elliptic integral expressions that are correct, but not very useful.
May
2
comment Why are there four independent solutions of Mathieu equation instead of two?
With the Bessels, there is a similar situation to that described by @mickep; you have the two kinds $J_\nu(z)$ and $Y_\nu(z)$, and then you have the Hankel functions which are complex combinations of the two Bessels.
May
2
comment Why does the midpoint method have error $O(h^2)$
You might want to look at the discussion by Hairer/Nørsett/Wanner.
May
2
comment Comprehensive compilation of conic section formulae
@MvG, if memory serves, what I obtained the last time I looked into this was just as horrendous; thanks, nevertheless.
May
2
comment How to reconstruct a symmetric matrix given the eigenvalues and eigenvectors.
Normalize the eigenvectors so that they're mutually orthogonal, and assemble them in a matrix $\mathbf V$. Then with the diagonal matrix of eigenvalues $\Lambda$, form $\mathbf V\Lambda\mathbf V^\top$. I omitted a few details that you should fill in.
May
1
comment Usage of integration representation
"In numerical calculation?" - sometimes, yes. As an example, one of the integral representations for the Bessel function of the first kind lends itself to a very useful method for numerical evaluation using the trapezoidal rule.