Principles of Mathematics in Operations Research

182 13 Power Series and Special Functions

x = 0 => L{1) — 0. Thus, we have

rv dx

L(y) = — = logy-

Ji x

Let u — E(x), v = E(y);

L{uv) = L{E{x)E(y)) = L{E{x + y))=x + y = L(u) + L(v).

We also have logx —> +oo as x —> +oo and logs —>• —oo as x —> 0. Moreover,

xn = E(nL(x)), x G R+; n, m G N, x™ = £ ( — L(x) J

xa = £(aL(x)) = eQlogx, Va € Q.

One can define xa, for any real a and any x > 0 by using continuity and

monotonicity of E and L.

(xa)' = E{aL{x))- = ax"-^1

One more property of log x is

lim x~a logx = 0, Va > 0.

x—»-f oo

13.5 Trigonometric Functions

Let us define

C(x) = \{E(ix) + E(-ix)}, S(x) = ^[E(ix) - E(-ix)}.

z zz

By the definition of £(z), we know E(z) = £(2). Then, C(x), S(x) G R, x G
R. Furthermore,
£(ix) = C(x)+iS(x).

Thus, C(x), 5(x) are real and imaginary parts of E(ix) if x G R. We have
also
|£(ix)|^2 = £(ix)£(ix) = E(ix)E{-ix) = E(0) = 1.

so that
\E(ix)\ = 1, x G R.

Moreover,

C(0) = 1, 5(0) = 0; and C"(x) = -S(x), S'(x) = C(x)