Mathematics of Physics and Engineering

(^214) Power Series
Euler's formula
eie = cos6 + ism6, (4.3.14)
and the relations
eiz — e~*z eiz 4- e~iz
sinz= — , cosz=. (4.3.15) 2i 2
We use the Euler formula to evaluate the complex exponentials: it fol-
lows from (4.3.14) that, for z = x + iy,
ez = ex(cosy + isiny), \ez\ — ex. (4.3.16)
Relation (4.3.16) shows that, as a function of complex variable, ez can take
all complex values except zero and is a periodic function with period 2m.
Similarly, (4.3.15) implies that sinz and cosz can take all complex values.
In particular, the familiar inequalities ex > 0, |sinx| < 1, |cosa;| < 1 that
are true for real x, no longer hold in the complex domain.
EXERCISE 4.3.13.B (a) Verify that the mapping defined by the exponential
function f(z) = ez is conformal everywhere in the complex plane. (b)Find
the image of the set {z : Uz > 0, 0 < Sz < 7r} under this mapping.
Two other related functions are the hyperbolic sine sinh and
hyperbolic cosine cosh:
ez — e~z ez -- e~z
sinhz = , coshz =. (4.3.17)
EXERCISE 4.3.14.B (a) Verify that
z2k+l z2k
sinh z — > 7— -77, cosh z = > -^rm •
(b) Verify that
sin(a; + iy) = sin x cosh y + i cos x sinh y, sinh(,z) = —i sin(iz),
cos(a: + iy) = cos a; coshy — isinx sinh y, cosh(z) = cos(iz).
The natural logarithm In z of z ^= 0 is defined as the number whose
exponential is equal to z. If a = In z, then, because of (4.3.14), a+2ni is also
a natural logarithm of z. In other words, In z has infinitely many values,
and, since z — \z\elaTe^z\ we have lnz = ln|z| +iaxg(z). The principal
value of the natural logarithm is defined by Ln z = In \z\ + i Arg(z), which