306.
(a) w = *Vz
(b) w = * y'r-a(_,-z---z-1.,.--:)(,--z ---z....,2 )-. -. ·-..,.( z---zs-:-) ( z; f z j, i f j)
(c) w =*Vaz - z1 z -z2 · .. z -Z6) (z; f Zj,i f j)
(d) w = *^3 a(z - z1)(z - z2)(z - za) (z; f Zj,i f j)
(e) W = *v'Z"=l + *~Z + 1
(f) w = log[(z - a)(z - b)]
z-a(g) w = log z _ b (a f b)
- Show that I Arcsinzl < %7r for izl < 1.
Chapter 5- Discuss w = arc sec z and construct the Riemann surface for this
function. Same problem for arccscz, i:;ech-^1 z, and csch-^1 z. - Suppose that lzl = 1 but z f ±i. Show that
Re( arc tan z) = ( n ±^1 / 4 )7r ( n an integer)
- Find all solutions of each of the following equations.
(a) sin z = 3 (b) cos z = i.
(c) sinz = J2 + cosz (d) tanz = ~
(e) sinhz = 2 (f) tanhz = -i. - Let w = loga z be the inverse of z =aw= ewLoga(a f 0, 1). Show that
1
logz
oga z = Log aDefine the principal logarithm to the base e so that Log 1 = 27ri
(instead of Log 1 = 0). Then
1 i
log 1 z = -
2argz - - ln lzl
7r 27r
so that log 1 z = 2 1,. argz whenever lzl = 1.- Discuss the mapping defined by w = z + logz.
- Find the values of each of the following powers.
(a) 3i (b) ii (c) (-1)-i
(d)(4+3i)^1 -i (e)(-4)v'2 (f)(l+i)^1 -i
- The Gudermannian of z is defined by
w = gdz = 2Arctanez -1/ 2 7r
Show that:
(a) z = gd-^1 w = Log tan^1 / 2 (w +^1 / 2 7r)
(b) sinhz = tanw
(c) coshz = secw