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How do I invert (revert) the following power series?

resolvent's picture
Posted on 2008-11-30 14:43 By resolvent (Maple Rating 2 94)
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Let w =f(z) = sum of z^(k+a) / (k + a)

where k= 0 to infinity and a is a nonzero parameter.

I need to find the inverse of this series, z = g(w). The powseries examples in Maple Help don't help. They don't work on my example, with a symbolic variable, a, stuck in there.  I hope that if I see about 7 or 8 terms of the inversion, I will get the general pattern.  I have tried to compute the inverse directly from the Lagrange Inversion Formula, but the complexity always grows too quickly for me to complete the solution, no matter which shortcut I try to take.

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Robert Israel's picture

inversion

Comment on 2008-11-30 15:56 from Robert Israel (Maple Rating 4 1555) 
> f := z^a * sum(z^k/(k+a), k=0..infinity);

= z^a*LerchPhi(z,1,a)

> R:= solve(f=w,z);

= RootOf(-_Z^a*LerchPhi(_Z,1,a)+w)

>  S := series(R, w, 4);

This result has the order-0 term RootOf(LerchPhi(_Z,1,a)).   I would guess, however, that (if a > 0) you would want the order-0 term to be 0,

and to get a series in powers of w^(1/a).  Let's call that s, and write the equation as

 

> eq := z * LerchPhi(z,1,a)^(1/a) = s;

= z*LerchPhi(z,1,a)^(1/a) = s

>  series(solve(eq,z), s, 8);

a^(1/a)*s+(-a^(1/a)/(a^(-1/a))/(a+1))*s^2+(-1/2*a^(2/a)*(a^2-a-4)/(a^(-1/a))/(a+2)/(a+1)^2)*s^3+(-1/3*a^(3/a)*(a^4-a^3-10*a^2+4*a+18)/(a^(-1/a))/(a+1)^3/(a+2)/(a+3))*s^4+(-1/24*a^(4/a)*(6*a^7+11*a^6-119*a^5-235*a^4+529*a^3+1016*a^2-632*a-1152)/(a^(-1/a))/(a+1)^4/(a+3)/(a+2)^2/(a+4))*s^5+(-1/10*a^(5/a)*(2*a^9+7*a^8-57*a^7-203*a^6+379*a^5+1378*a^4-888*a^3-3222*a^2+684*a+2400)/(a^(-1/a))/(a+4)/(a+3)/(a+5)/(a+1)^5/(a+2)^2)*s^6+(-1/720*a^(6/a)*(-45078*a^10+9049664*a^3-4942098*a^4-140531*a^9-4936896*a+120*a^13+388306*a^6+252156*a^8+1555497*a^7+159*a^11-6220800+1322*a^12+10065072*a^2-5856333*a^5)/(a^(-1/a))/(a+5)/(a+4)/(a+1)^6/(a+2)^3/(a+3)^2/(a+6))*s^7+O(s^8)

 

resolvent's picture

LerchPhi

Comment on 2008-11-30 18:36 from resolvent (Maple Rating 2 94)

Wow! I had never heard of the LerchPhi function before. I knew in my heart that SOMEBODY must have given this function a special name. Thank you.

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