Re: [math-fun] Slanted coverage
Further playing with the big hammer dA = dt du |V_t x V_u| gives the area of a 1x1xc spheroid as 2 pi (1 + c^2 asec(c)/sqrt(c^2-1)), with the axaxb general case following easily. This agrees with Mathworld's prolate (c>1) formula, but for oblate, Weisstein gives the logarithmic equivalent that avoids imaginary/imaginary. Something must be wrong with our notation if we need to switch at c = 1. The function there is smooth as a marble!-) A = 2 pi (1 + c 2F1[1,1,3/2,1/2-1/2c]) . (Rotsa ruck coaxing this out of Mma. Or even checking it.) A simpler example might be Plot[ArcCos[x]^2,{x,-1,5}], where the plotter needed to call a complex-valued function to get real results. Should we put acos^2 on pocket calculators? --rwg Joerg: The images are now (again, for me) all visible. Thanks for fixing this. The thanks go to Neil for rescuing the .nb and creating the working .html with his Windows 7 Machine.
On Sun, Apr 25, 2010 at 8:53 PM, Bill Gosper <billgosper@gmail.com> wrote:
Further playing with the big hammer
dA = dt du |V_t x V_u|
gives the area of a 1x1xc spheroid as
2 pi (1 + c^2 asec(c)/sqrt(c^2-1)),
with the axaxb general case following easily. This agrees with Mathworld's prolate (c>1) formula, but for oblate, Weisstein gives the logarithmic equivalent that avoids imaginary/imaginary.
Oops, I was working with an ancient offline copy. Web version now has the general 2F1 formula in the Spheroid entry, but a different 2F1! An interesting quadratic transformation congrues them. Something must
be wrong with our notation if we need to switch at c = 1. The function there is smooth as a marble!-)
A = 2 pi (1 + c 2F1[1,1,3/2,1/2-1/2c]) .
(Rotsa ruck coaxing this out of Mma. Or even checking it.)
A simpler example might be Plot[ArcCos[x]^2,{x,-1,5}], where the plotter needed to call a complex-valued function to get real results. Should we put acos^2 on pocket calculators? --rwg
Exercise: Sketch the graph of asin(sqrt(sin(x)))^2. Now compare with Mathemaplema. For a mild start, plot the derivative. --rwg
Joerg: The images are now (again, for me) all visible. Thanks for fixing this.
The thanks go to Neil for rescuing the .nb and creating the working .html with his Windows 7 Machine.
A simpler example might be Plot[ArcCos[x]^2,{x,-1,5}], where the plotter
needed to call a complex-valued function to get real results. Should we put acos^2 on pocket calculators? --rwg
We should put complex numbers on pocket calculators.
Exercise: Sketch the graph of asin(sqrt(sin(x)))^2. Now compare with Mathemaplema. For a mild start, plot the derivative. --rwg
Shifting by pi/2, asin(sqrt(cos(x)))^2 is an even, periodic fn with a
cosine all-positive Fourier series, with a5F4[p,...] constant term and 4F3 higher terms (http://tweedledum.com/rwg/asincos.png). Mma was little
help:
SeriesCoefficient flunks ArcSin^2, and after you tell it, Assuming[k > 0 && n > 0 && k \[Element] Integers && n \[Element] Integers, Integrate[Cos[k*x]*Cos[x]^n, {x, -Pi, Pi}]] is a mess including Gamma[-k]. --rwg
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Bill Gosper