18.14 HINTS AND ANSWERS
18.15 (a) Show that the ratio of two definitions based onmandn,withm>n>−Rez,
is unity, independent of the actual values ofmandn.
(b) Consider the limit asz→−mof (z+m)z!, with the definition ofz!basedon
nwheren>m.
18.17 Express the integrand in partial fractions and use∫ J,asgiven,andJ′ =
∞
0 exp[−iu(k−ia)]duto expressIas the sum of two double integral expressions.
Reduce them using the standard Gaussian integral, and then make a change of
variable 2v=u+2a.
18.19 (b) Using the representation
M(a, b;z)=Γ(b)
Γ(b−a)Γ(a)∫ 1
0eztta−^1 (1−t)b−a−^1 dtallows the equality to be established, without actual integration, by changing the
integration variable tos=1−t.
18.21 Calculatey′(x)andy′′(x) and then eliminatex−^1 e−xto obtainxy′′+(n+1+x)y′+
ny=0;M(n, n+1;−x). Comparing the expansion of the hypergeometric series
with the result of term by term integration of the expansion of the integrand
shows thatA=n.
18.23 (a) If the dummy variable in the incomplete gamma function ist, make the change
of variabley=+
√
t.Nowchooseaso that 2(a−1) + 1 = 0; erf(x)=P(^12 ,x^2 ).
(b) Change the integration variableuin the standard representation of the RHS
tos,givenbyu=^12√
π(1−i)s, and note that (1−i)^2 =− 2 i.A=(1+i)/2. From
part (a),C(x)+iS(x)=^12 (1 +i)P(^12 ,−^12 πi x^2 ).