Elementary Functions- Show that the vertical strip
S = {z: z = x + iy, -a:::; x:::; a, 0 <a< %7r}
is mapped by w = tan('1rz/4a) onto the disk lwl :::; 1.
28519. Given w EC - {(-ioo, -i] U [i, ioo )}, construct geometrically the arcs
Ca, Cb intersecting at w (Fig. 5.25). Hint: Note that Ca intersects
Cb orthogonally..- Show that f(z) = cosh(ax +by), where z = x + iy, and a and b are
arbitrary complex constants, satisfies the functional equation
f(z + z') + f(z - z') = 2f(z)f(z') - Discuss the mapping defined by w = sec z.
- Show that the roots of the equation
(z+a)ez=z-a
where a is real, are real or pure imaginary. - A complex number e is said to be algebraic if it is a root of some
polynomial equation ao + aiz + · · · + anzn = 0 (an -=f. O,n;?: 1) with
integral coefficients. Otherwise, e is said to be transcendental. Hermite
proved in 1873 that e is transcendental, and Lindemann in 1882 proved
that 7r is transcendental. Prove that the transcendence of both e and 7r
follows from Lindemann's theorem: If for k = 1, 2, ... , n the numbers
bk and Ck are algebraic with Ck -=f. Cj fork -=f. j, and bk -=f. 0 for all k, then
In the remainder of this chapter we discuss some elementary examples of
multiple-valued functions, namely, the inverses of some of the functions
studied in preceding sections.
5.20 THE FUNCTION w = * .ifZ. INTRODUCTION OF
THE RIEMANN SURFACES
If n is a positive integer greater than 1, and if z -=f. 0, oo there are n distinct
complex numbers w such that
which are the nth roots of z (Section 1.12). If we let z = rei^6 , w = pei.P,
we have
p= \ii
() 27r
V;=-+k-
n nand