## Πέμπτη 20 Ιουνίου 2019

### Logarithm of a negative number

It is often said that we cannot take logarithm of a negative number. But wait! Suppose we can use complex numbers.

### Euler formula

We begin with the Euler formula:
and put q = p, we get :
Therefore,
ln (– 1) = ip
If a > 0, then
ln (– a) = ln [a(– 1)] = ln a + ln (– 1) = ln a + ip
So, ln (– 2) = ln 2 + ip » 0.69315 + 3.1416 i
and

### Logarithm of a complex number

By putting q = p/2 in the Euler formula, we get
Therefore

If z = b i, which is a purely imaginary number,
then

Finally, for the polar form of a complex number,
,
we get