1550251515-Classical_Complex_Analysis__Gonzalez_

730

while

lim J fo(z)dz = i7rVC

6-+0

'"Yo '+

lim J fo(z) dz= i7r(-VC)

6-+0

-rt

Chapter9

since limz-+O zf 0 (z) equals VG or -VG depending on whether z is re-

stricted to the lower or upper indentation. Thus if we let e -+ 0 in (9.11-65)

and combine similar terms, we get

~ ~ 2 [[' fo(x)& + l fo(x)&l + / fo(z)dz+ [ fo(z)dz

'"Yo+ '"Yo

Finally, letting 6 -+ 0, we find that t

l

b 1 7rB

(PV) -V Ax^2 + 2Bx + C dx = /7

a x v-A

(9.11-66)

EXERCISES 9.7

Show that:

1 {

00

lnx dx = ._ 7r^2 ../2

· lo x^4 +1 16

2. f 00 (ln x )^2 dx = 37r^3 ../2
   lo x^4 + 1 64

1

(^00) ln(x (^2) + 1)
*3. 2
1
dx = 7r ln 2, and deduce that
0 x +
r<l/2)7r r<l/2)7r
lo lnsinBdB =lo lncosBdB = -^1 / 2 7rln2
4

1

1

ln(x + l/x) d = ~ 1 2

• 2 1 x 2?Tn
  0 x +

1

(^00) lnx dx 7r

5. ·( 2 2 ) 2 = - 3 (Ina - 1), a > 0

0 x +a 4a

tThe results in formulas (9.11-64) and (9.11-66) are contained in a joint paper
by E. Badell and the author (1], published in 1942. Reproduced in part by V. I.
Smirnov (22] in 1957.