Geometry/Exponentiation Two Complex Numbers
Dear Prof Azeem
Can we compute exponentiation of two complex numbers?
Example- 1+2i and 2+3i
I.e 1+2i raised to power of 2+3i
Complex exponentiation is doable, but it can get messy very quickly.
Let's assume we have complex numbers w and z. I will assume throughout that we have an appropriately defined logarithm so that w^z is unambiguous.
The cleanest way to do this is using polar coordinates. (If you are not familiar with polar coordinates, ask a follow-up.) Let r and x be such that w=re^(ix). Taking the logarithm of both sides, we obtain,
log(w) = log(r) + ix
Our concern is w^z, which is equal to e^[(z)log(w)] by a change of base. We can now substitute in the result from above for log(w).
w^z = e^[(z)log(w)] = e^[(z)(log(r)+ix)]
Note that the complex number z was not touched throughout.
To compute your example, we need only compute the modulus and the argument of w, and substitute.
w=1+2i ; z=2+3i
Substituting it all in,
Thanks for asking,