Sequences, Series, and Special FunctionsHence,z = Log () 1 + w = w - l2 -w + -l3 w - -^14 w + · · ·
2 3 4
which converges for lwl < 1.8.5 FURTHER SERIES EXPANSIONS. THE SYMBOLIC
METHOD. BERNOULLI AND EULER NUMBERS545(8.4-12)The derivation of the power series expansion of certain trigonometric and
hyperbolic functions is greatly facilitated by using a device introduced by
E. Lucas. This symbolic method may be described as follows: Consider a
power series written in the form
(8.5-1)If we write instead
1 a^2 2 an n az
l+a z+ -z +···+ -z +···=e
2! n!
(8.5-2)then (8.5-1) may be taken as the formal expansion of eaz, provided that
each an is interpreted as equivalent to an. With this convention, we shall
write (8.5-2) with the equal sign ( =) replaced by ='=:
e = + az ·^1 a a2^2 an n
1 z + -z 2! + · · · + -z n! + · · ·
Let
bz • b b2 2 bn n
e =1+ 1z+-z 2! +···+ -z n! +···Then, for any two complex constants a, (3, we have
b. aa2 + f3b (^2 2)
aeaz + (3e z = (a + (3) + ( aa1 + f3b1 )z + 1 z + · · ·
2.Also, we have
eazebz=e(a+b)z=l+(a+b)lz+ (a+b)2 z2+···+ (a+b)n zn+···
2! n! ·so that
e e = + aaz bz. l ( b ) a2 + 2a1 b1 + b2^2
1 + 1 z + 21 z + · · ·
an + nan-1 b1 + · · · + bn n
- n! z +···