Add sinh(tanh(x)) and other similar functions.

[emsr]

Work on double exp integrals and recent fast math work in gcc indicates that sinh(tanh(x)), cosh(tanh(x)), sinh(atanh(x)), and cosh(atanh(x)). I'm not interested in optimizing but in accuracy.

[CaptainSifff]

Hi, do you have a pointer on what is needed for the hyperbolic functions? Are you interested in reliable series for e.g. sinh(tanh(x)) ?

[emsr]

On 6/21/19 9:01 AM, CaptainSifff wrote:

Hi, do you have a pointer on what is needed for the hyperbolic functions? Are you interested in reliable series for e.g. sinh(tanh(x)) ?

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Yes, I would like a good series for sinh(tanh(x)).

Maybe for sinh(a*tanh(x)) since these are used in double exponential quadrature rules (exp a = pi/2).

Thanks!

[CaptainSifff]

And directly evaluating that function is not fast enough? or lacks precision for certain values since some overflow/underflow occurs?

[CaptainSifff]

Hi Ed, I've got something to play for you... We consider the functions C(x) = Cosh(a Tanh(x)) and S(x) = Sinh(a Tanh(x)) and consider the behaviour for large x. Denoting real infinity as "inf" We find C(inf) = Cosh(a) and S(inf) = Sinh(a) I think I can show that the large-x behaviour is given by C(x) ~ C(inf) F(a,x) and S(x) ~ S(inf) F(a,x) with the same function F(a,x). Now we have F(a,x) = y(a,x) K_1(y(a,x)) with y(a,x) = 2 a exp(-x) and K_1 denoting the modified Bessel function of the second kind (http://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html) Since y(a,x) becomes exponentially small for large x we can use any Series around x=0 for K_1. To give an approximation to first order that only relies on elementary functions we have: F(a,x) ~ 1 + a^2 exp(-2x) (-1+2 \gamma + 2 ln(a) -x) + ... Here \gamma is the Euler-Mascheroni constant(https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant) I get good agreement in Mathematica between C(x) evaluated in Mathematica for a=1 and x>20. If you have questions or my notation is unclear, just ask!

[CaptainSifff]

Forget what I have previously written... I found a "minor" bug in my calculation that makes everything far simpler.... To begin again: We consider the functions C(x) = Cosh(a Tanh(x)) and S(x) = Sinh(a Tanh(x)) and consider the behaviour for large x. Denoting real infinity as "inf" We find C(inf) = Cosh(a) and S(inf) = Sinh(a) I think I can show that the large-x behaviour is given by C(x) ~ C(inf) F(a,x) and S(x) ~ S(inf) F(a,x) with the same function F(a,x). Now we have F(a,x) = Cosh(2 a exp(-2 x)) Since exp(-2 x) becomes exponentially small for large x we can use any Series around x=0 for Cosh(x). To give an approximation to second order in exp(-2x) that only relies on elementary functions we have: F(a,x) ~ 1 + 2 a^2 exp(-4x) (more terms could be obtained from the Taylor series of cosh(x) but they are probably not required...) I get good agreement in Mathematica of C(x) evaluated in Mathematica for a=1 and x>20. If you have questions or my notation is unclear, just ask!

[emsr]

Very nice. I'm going to play with this myself. I'm not sure what I'd call these ;-) but we'll think of something.

[CaptainSifff]

the most unimaginative I could come up with is: C_a(x) := cosh(a tanh(x)) S_a(x) := sinh( a tanh(x))