Mathematica Stack Exchange is a question and answer site for users of Wolfram Mathematica. It only takes a minute to sign up.
Sign up to join this community
f[m_, z_] := (1/m)*Sum[Exp[z*Exp[2*Pi*I/m]^k], {k, 0, m - 1}]
g[t_] := 1/(2 - f[5, t^(1/5)])
Series[g[t], {t, 0, 10}]
When I tried to compute this on Wolfram Programming Cloud I got the message: "The Wolfram Engine has been terminated because the evaluation time limit was reached."
Perhaps the function g can be rewritten in a simpler form (a sum of trigonometric terms)?
It should be noted that the Mittag-Leffler function is built-in:
f[m_, z_] := MittagLefflerE[m, z^m]
g[t_] := 1/(2 - f[5, t^(1/5)])
Series[g[t], {t, 0, 10}]
which yields the same series as in the Wizard's answer. If you are using an older version that does not yet support MittagLefflerE[], here is an alternative:
f[m_, z_] := HypergeometricPFQ[{}, Range[m - 1]/m, (z/m)^m]
g[t_] := 1/(2 - f[5, t^(1/5)])
Series[g[t], {t, 0, 10}]
My best guess as to what you want:
f[m_, z_] := (1/m)*Sum[Exp[z*Exp[2*Pi*I/m]^k], {k, 0, m - 1}]
g[t_] := 1/(2 - f[5, t^(1/5)]);
Series[ExpToTrig @ g[t], {t, 0, 10}]
1 + t/120 + (253 t^2)/3628800 + (762763 t^3)/1307674368000 + ( 43173223 t^4)/8846916393369600 + (633287284180541 t^5)/15511210043330985984000000 + ( 633594892177711781 t^6)/1854915103581755654799360000000 + ( 29529277377602939454694793 t^7)/10333147966386144929666651337523200000000 + ( 118226228593807528558241820049 t^8)/4944941110593319602094613755127975116800000000 + \ (1259341493633888212897976517963115369 t^9)/ 6295906361341062871682271657666195529704407040000000000 + \ (107604770495341349966501435996477524098581 t^10)/ 64300408460281983178885006693583020812637720018944000000000000
(I used Normal on the output for the sake of copying.)
isupposed to be imaginary i? If so it needs to beI(capital letter). $\endgroup$