You are here:

# Advanced Math/Compute exponentiation of two complex numbers.

Question
Dear Prof Scott

Can we compute the exponentiation of two complex numbers?

Example - 1+2i and 2+3i
I.e 1+2i raised to power of 2+3i

Thanks
Prashant

Yes, it can be done.  They have found that complex numbers allow all operations to be done.
Note that with positive integers were the first thing thought of, since people count.
Division leads to the development of fractions.  Subtraction leads to the development of negative numbers.  Square roots of positive numbers lead to the development of more real numbers than just fractional numbers.  Square roots of negatives lead to the development of imaginary numbers.  Using a xy axis, with y as the imaginary, complex numbers can be formed.  This is the final set to use, because any operation on a complex number results in another complex number.

When exponentials are involved, the answer is found in polar coordinates.
It is known that if x = a + bi, it can be converted to polar coordinates by finding numbers such that Ce^(iΘ) = C*[cos(Θ) + i*sin(Θ)].

To do this, if the number is a + bi, we need to find C²=a²+b².

With 1 + 2i, C² = 1² + 2² = 1 + 4 = 5.  This means that C = √5.
This converts the number to √5[√5/5 + 2(√5/5)i].

The angle Θ then gives us sin(Θ) = 2*√5/5 and cos(Θ) = √5/5.

The number is (√5)e^(iΘ).

When we raise this to the power 2+3i, it is  [(√5)e^(iΘ)]^(2+3i).
This is the same as  (√5){[e^(iΘ)]^2}{[e^(iΘ)]^3i}.

When we have (a^b)^c, it is the same as a^(bc).
That gives us (√5){e^(2iΘ)}{(√5)e^(-3Θ)}.

Using  Ce^(iΘ) = C*[cos(Θ) + i*sin(Θ)], that number can be found.

Now there might be a pun here, but I hope this isn't too complex ...
after all, some of this is purely imaginary ...

Volunteer

#### Scott A Wilson

##### Expertise

I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can also say that I broke 5 minutes for a mile, which is over 12 mph, but is that relevant?

##### Experience

Experience in the area; I have tutored people in the above areas of mathematics for over two years in AllExperts.com. I have tutored people here and there in mathematics since before I received a BS degree back in 1984. In just two more years, I received an MS degree as well, but more on that later. I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman. I tutored at Mathnasium for well over a year. I worked at The Boeing Company for over 5 years. I received an MS degreee in Mathematics from Oregon State Univeristy. The classes I took were over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more. I graduated with honors in both my BS and MS degrees. Past/Present Clients: College Students at Oregon State University, various math people since college, over 7,500 people on the PC from the US and rest the world.

Publications
My master's paper was published in the OSU journal. The subject of it was Numerical Analysis used in shock waves and rarefaction fans. It dealt with discontinuities that arose over time. They were solved using the Leap Frog method. That method was used and improvements of it were shown. The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.

Education/Credentials
Master of Science at OSU with high honors in mathematics. Bachelor of Science at OSU with high honors in mathematical sciences. This degree involved mathematics, statistics, and computer science. I also took sophmore level physics and chemistry while I was attending college. On the side I took raquetball, but that's still not relevant.

Awards and Honors
I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

Past/Present Clients
My clients have been students at OSU, people who live nearby, friends with math questions, and several people every day on the PC. I would guess that you are probably going to be one more.