**Complex Exponents With Positive Real Bases**

If *b* is a positive real number, and *z* is any complex number, the power *b**z* is defined as *e**z*·ln(*b*), where *x* = ln(*b*) is the unique real solution to the equation *e**x* = *b*. So the same method working for real exponents also works for complex exponents.

For example:

- 2
*i*=*e**i*·ln(2) = cos(ln(2)) +*i*·sin(ln(2)) ≈ 0.76924 + 0.63896*i* *e**i*≈ 0.54030 + 0.84147*i*- 10
*i*≈ −0.66820 + 0.74398*i* - (
*e*2π)*i*≈ 535.49*i*≈ 1

The identity is not generally valid for complex powers. A simple counterexample is given by:

The identity is, however, valid when is a real number, and also when is an integer.

Read more about this topic: Complex Numbers Exponential, Complex Exponents With Positive Real Bases

### Famous quotes containing the words bases, real, complex and/or positive:

“The *bases* for historical knowledge are not empirical facts but written texts, even if these texts masquerade in the guise of wars or revolutions.”

—Paul Deman (1919–1983)

“A decent chap, a *real* good sort,

Straight as a die, one of the best,

A brick, a trump, a proper sport,

Head and shoulders above the rest;

How many lives would have been duller

Had he not been here below?

Here’s to the whitest man I know

Though white is not my favourite colour.”

—Philip Larkin (1922–1986)

“Power is not an institution, and not a structure; neither is it a certain strength we are endowed with; it is the name that one attributes to a *complex* strategical situation in a particular society.”

—Michel Foucault (1926–1984)

“Nurturing competence, the food of self-esteem, comes from acknowledging and appreciating the *positive* contributions your children make. By catching our kids doing things right, we bring out the good that is already there.”

—Stephanie Martson (20th century)