Complex Numbers Exponential - Complex Exponents With Positive Real Bases - Complex Exponents With Positive Real Bases

Complex Exponents With Positive Real Bases

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

For example:

2i = e i·ln(2) = cos(ln(2)) + i·sin(ln(2)) ≈ 0.76924 + 0.63896i
ei ≈ 0.54030 + 0.84147i
10i ≈ −0.66820 + 0.74398i
(e2π)i ≈ 535.49i ≈ 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

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