19/04/03 20:40:39.38 /TkvX91f.net
>>817
sin(x) = {exp(ix) - exp(-ix)} /2i,
exp(x)sin(x) = {exp((1+i)x) - exp((1-i)x)} /2i,
exp(x)sin(x) の x^p の係数は
f_p = (1/p!) {(1+i)^p - (1-i)^p} /2i
= (1/p!)(√2)^p {exp(iπ/4)^p - exp(-iπ/4)^p} /2i
= (1/p!)(√2)^p {exp(i(pπ/4)) - exp(-i(pπ/4))} /2i
= (1/p!)(√2)^p sin(pπ/4),
exp(x) sin(x) = x + x^2 + (1/3)x^3 +0･x^4 - (1/30)x^5 - (1/90)x^6 - (1/630)x^7 -0･x^8 + (1/22680)x^9 + (1/113400)x^10 + (1/1247400)x^11 + 0･x^12 - ････
1/(1-x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + ････
ゆえ
f(x)/(1-x) の x^pの係数は Σ[k=0,p] f_k
exp(x)sin(x)/(1-x) = x + 2x^2 + (7/3)(x^3 +x^4)
+ (23/10)x^5 + (103/45)x^6 + (1441/630)(x^7 +x^8)
+ (7411/3240)x^9 + (43231/18900)x^10 + (2853247/1247400)(x^11 +x^12)
+ (4046423/1769040)x^13 + (778936427/340540200)x^14 + (23368092809/10216206000)(x^15 +x^16)
+ (42374141627/18525386880)x^17 + (14301272799113/6252318072000)x^18 + (8360744097943/3655201334400)(x^19 +x^20)
+ ････