1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 549

The numbers E 2 n [or the ( -1 r E 2 n] are called the Euler numbers. They

were introduced by L. Euler and later called Euler numbers by H. F. Scherk
in 1825. They are all odd integers and terminating with either 1 or 5. To
Scherk is also due the formula

E 2 = 1 [l ~ (2n + 1) 2 2k+i( 2 2k-1 l)B l

n 2n + 1 LJ 2k^2 k

. k=l

expressing the Euler numbers in terms of the Bernoulli numbers.
Returning to (8.5-12), we have

sec h z=^1 + ~ ~ (E2n 2n

2 n)!z ,

(8.5-13)

and replacing z by iz, we obtain

sec z = 1 + ~(-l)n (~~)! z2n, (8.5-14)

Extensive tables of the Bernoulli and Euler numbers have been published
(see, e.g., [14]). It is know that the numbers (-1r-^1 B2n and (-l)n E2n
are all positive (See Section 9.12, Examples 3 and 5; see also papers [4]
and [26]).

Exercises 8.1

1. Expand each of the following functions in powers of z and find the
   radius of convergence of the resulting series.
   (a) sinhz (b) coshz
   1 z
   (c) 2+z (d) l+z
   (e) sin^2 z (f) cosh^2 z
   (g) Log(l - z) (h) Sinh-^1 z
   (i) Tanh-^1 z (j) (1 + z)^312


2. Expand each of the following in powers of z - 1 and find the radius of
   convergence of the resulting series.
   1 z
   (a) - (b) -

z 1 +z

(c) ez (d) Logz

(e) Arctanz (f) sinz

(g) 1 (h) z1/2 (l1/2 = l)

(l+z)^2.

3. Expand f(z) = z^1 /^2 in powers of z-(-1 +i) and determine the region
   of validity of the expansion.