1550251515-Classical_Complex_Analysis__Gonzalez_

642 Chapter^8

defining series converges absolutely (by th~ ratio test) for all values of

z E C, and so cI>( a, c; z) is an entire function of z. This function plays a

rather important role in analysis due to the fact that a good number of

special functions are particular cases of cI>( a, c; z ).

Examples

1.

oo n

<P(a,a;z) = L; = ez

n=O n.

2.

oo zn ez -1

g~(l, 2; z) = ~ (n + 1)! = -2-

3. E'irf z = r e-t2 dt = f (-lr z2n+l
   lo n=O n!(2n + 1)

~ (%)n ( 2)n ;r,. ( 1 3 2)

= z ~ n!(%)n -z = Z'i! 2' 2; -z

4.

l

z • 7rt2 ~ (-l)n ( 7r )2n+l Z4n+3

S(z) = sm - dt = L; - --

0 2 n=O (2n + 1)! 2 4n + 3

= ;i [cf> ( ~ ' ~ j 7r~Z2 ) - cf> ( ~ ' ~ j - 7r~2 ) ]

LOl()=(a:+l)k~ (-k)n n=(a:+l)k;r,.(-k +l·)

k z k'.. L.t n. '( a:+ 1) n z kl. 'i! 'a: 'z

";· n=O

5.

Theorem 8.52 Th~ confluent hypergeometric function cI>( a, c; z) has the
following properties:

dm (a)m

l. dzm cI>(a,c;z) = (c)m cI>(a+m,c+m;z), m = 1,2, ...

2. ccI>(a,c;z) = c<P(a -1,c; z) + zcI>(a, c + 1; z)


3. cI>(a,c;z)= ( r~c) ) f

1

eztta-^1 (l-t)c-a-^1 dt, Rec>Rea>O

r arc -a lo

4. cI>(a, c; z) = ez<J?(c - a,c;-z)


5. zcI>" + (c - z)cI>' - acI> = 0

Proofs (1) From (8.23-1) we get

d ( ) ~ (a)n n-1 ~ (a)n+i n

d cI> a, c; z = L.t ( _ l)'( ) z = L.t '( ) z

z n=l n. c n n=O n. c n+I