1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 633

when Re a, Re b # 0, -1, -2, ....

5. Show that r(z) = I'(z). Also, prove that

7r

r(z)r(-z) = --. -

ZS1Il7rZ

and deduce that

1r(iy)l

2

=. :

ysm 1rY

6. From
   7r

r(z)I'(l - z) = -. -

SIU 'TrZ

show that

1r(%+iy)l

2

= cos ~ 1rY

and hence

y-+±oo lim r(1/2 + iy) =^0

7. Show that

B-(n n) = J?fI'(n) = 21-2n B(n,1/2)

' 22n-1r(n + %)

8. Prove the following.
   (a) B(z, ()B(z + (, w) = B((, w)B(( + w, z), z, (, w # 0, -1, -2, ...

(b) B(z, n + 1) = n!/(z)n+i, z # 0, -1, -2, ...

9. Evaluate.
   (a) fl dx
   lo )1 - x^4

(b) fl dx

(c) 1

1

{!t2(1-t)dt

r12

(e) lo ~de

lo )1 - x^114

[1 x^4 dx

(d) lo (l+x)^9

r12

(f) lo (sin20)

1

1

4

d0

10. Show that

(1 - tttz-l dt = n. ,

1

1 '

o z(z + 1) · · · (z + n)

11. (a) Show that the length of the lemniscate

r^2 = a^2 cos20

is given by L = a0fI'(1/ 4 )/r(3/4) ~ 5.24a.

(b) Find the area bounded by the oval

(1 + x^2 )y^2 = 1 - x^2

Rez > 0