sin(arcsinh(x)) and cos(arcsinh(x): Taylor series?

Richard L. Peterson

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Feb 5, 2010, 3:22:20 PM2/5/10

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I would be intrigued to know the Taylor
series for these functions. Best wishes,
Rich Peterson

Gottfried Helms

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Feb 5, 2010, 3:40:44 PM2/5/10

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Pari/GP says:

sin(asinh(x))
%1778 = x - 1/3*x^3 + 1/6*x^5 - 13/126*x^7 + 325/4536*x^9 - 2665/49896*x^11 + 12505/299376*x^13 - 212585/6286896*x^15 + 1413065/50295168*x^17 - 204894425/8600473728*x^19 + 7417178185/361219896576*x^21 - 1639196378885/91388633833728*x^23 + 17375481616181/1096663606004736*x^25 - 418348134297281/29609917362127872*x^27 + 5265416173051985/414538843069790208*x^29 - 443348041770977137/38552112405490489344*x^31 + O(x^33)

cos(asinh(x))
%1779 = 1 - 1/2*x^2 + 5/24*x^4 - 17/144*x^6 + 629/8064*x^8 - 8177/145152*x^10 + 825877/19160064*x^12 - 9211705/268240896*x^14 + 362941177/12875563008*x^16 - 5486816617/231760134144*x^18 + 356643080105/17613770194944*x^20 - 143013875122105/8137561830064128*x^22 + 69361729434220925/4491934130195398656*x^24 - 123143747333986073/8983868260390797312*x^26 + 83368316945108571421/6791804404855442767872*x^28 - 451338819323518817693/40750826429132656607232*x^30 + 406656276210490454741393/40424819817699595354374144*x^32 + O(x^34)

Gottfried Helms

Dave L. Renfro

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Feb 5, 2010, 3:59:08 PM2/5/10

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Richard L. Peterson wrote:

> I would be intrigued to know the Taylor
> series for these functions. Best wishes,

These seem to be close relatives to the Gudermannian function gd(x).
If you don't know anything about gd(x), see the following post:

http://groups.google.com/group/sci.math/msg/dfb992fe3d16fc49

Dave L. Renfro

Richard L. Peterson

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Feb 5, 2010, 3:56:47 PM2/5/10

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Vielen Dank!

Robert Israel

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Feb 5, 2010, 7:38:31 PM2/5/10

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In Maple 13:

> convert(sin(arcsinh(x)), FPS,x);

Sum(Product(1+2*j+2*j^2,j = 0 .. k)*(-2)^k/(2*k^2+2*k+1)/(2*k+1)!*x^(2*k+1),
k = 0 .. infinity)

> convert(cos(arcsinh(x)), FPS,x);

Sum((-1)^k/(4*k^2+1)*Product(4*j^2+1,j = 0 .. k)/(2*k)!*x^(2*k),k = 0 .. infinity)
--
Robert Israel isr...@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

Ray Vickson

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Feb 6, 2010, 3:21:00 PM2/6/10

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On Feb 5, 4:38 pm, Robert Israel
<isr...@math.MyUniversitysInitials.ca> wrote:

> "Richard L. Peterson" <rl_p...@yahoo.com> writes:
>
> > I would be intrigued to know the Taylor
> > series for these functions. Best wishes,
> > Rich Peterson
>
> In Maple 13:
>
> > convert(sin(arcsinh(x)), FPS,x);
>
> Sum(Product(1+2*j+2*j^2,j = 0 .. k)*(-2)^k/(2*k^2+2*k+1)/(2*k+1)!*x^(2*k+1),
> k = 0 .. infinity)
>
> > convert(cos(arcsinh(x)), FPS,x);
>
> Sum((-1)^k/(4*k^2+1)*Product(4*j^2+1,j = 0 .. k)/(2*k)!*x^(2*k),k = 0 .. infinity)

Do you know what type of algorithm Maple 13 would use to get this?

R.G. vickson

Robert Israel

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Feb 6, 2010, 10:53:18 PM2/6/10

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Ray Vickson <RGVi...@shaw.ca> writes:

> On Feb 5, 4:38=A0pm, Robert Israel

> <isr...@math.MyUniversitysInitials.ca> wrote:
> > "Richard L. Peterson" <rl_p...@yahoo.com> writes:
> >
> > > I would be intrigued to know the Taylor
> > > series for these functions. Best wishes,
> > > Rich Peterson
> >
> > In Maple 13:
> >
> > > convert(sin(arcsinh(x)), FPS,x);
> >

> > Sum(Product(1+2*j+2*j^2,j =3D 0 ..
> > k)*(-2)^k/(2*k^2+2*k+1)/(2*k+1)!*x^(2*=
> k+1),
> > =A0 =A0 =A0 k =3D 0 .. infinity)
> >
> > > convert(cos(arcsinh(x)), FPS,x);
> >
> > Sum((-1)^k/(4*k^2+1)*Product(4*j^2+1,j =3D 0 .. k)/(2*k)!*x^(2*k),k =3D
> > 0=

> .. infinity)
>
> Do you know what type of algorithm Maple 13 would use to get this?
>
> R.G. vickson

It appears to find a linear differential equation the function satisfies:
in the case of sin(arcsinh(x)) that's (1+x^2)*y'' + x*y' + y = 0.
From there you go to a recurrence for the Taylor coefficients.

Richard L. Peterson

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Feb 7, 2010, 8:05:34 PM2/7/10

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Thanks for everyone's thoughtful replies and
questions. Rich Peterson