Jacques Gélinas on Thu, 08 Aug 2019 22:56:00 +0200


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Re: MacLaurin expansion of even functions 

\ps 4
xis(s) = gamma(1+s/2)/Pi^(s/2)*(s-1)*zeta(s);

xis(1/2+I*(t+O(t^9)) )
0.49712 + 0.E-57*I*t - 0.011486*t^2 + 1.2745 E-57*I*t^3 + 0.00012345*t^4 - 1.5294 E-56*I*t^5 - 8.3236 E-7*t^6 + 0.E-55*I*t^7 + 3.9922 E-9*t^8 + O(t^9)

A really lazy way for this real case is
subst(real( xis(1/2+I*(t+O(t^9)) ) ), t,I*t)
0.49712 + 0.011486*t^2 + 0.00012345*t^4 + 8.3236 E-7*t^6 + 3.9922 E-9*t^8 + O(t^9)

but this is better, avoiding complex arithmetic,
Sereven(f)=my(V=Vec(f),v=variable(f));forstep(i=2,#V,2,V[i]=0);Ser(V,v)*t^valuation(f,v);

Sereven( xis(1/2+t+O(t^9)) )
0.49712 + 0.011486*t^2 + 0.00012345*t^4 + 8.3236 E-7*t^6 + 3.9922 E-9*t^8 + O(t^9)

Thanks,

Jacques Gélinas