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Geometry/Exponentiation Two Complex Numbers

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Question
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

Thanks
Prashant

Answer
Hi Prashant,

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

Modulus
r=a+b
r=(1)+(2)
r=5
r=√5

Argument
x=arctan(b/a)
x=arctan[(2)/(1)]
x=arctan(2)

Substituting it all in,

(1+2i)^(2+3i)=e^[(2+3i)log(√5)+i(arctan(2))]


Thanks for asking,
Azeem

Geometry

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Azeem Hussain

Expertise

I welcome your questions on algebra, 2D and 3D geometry, parabolic functions and conic sections, and any other mathematical queries you may have.

Experience

4 years as a drop-in and by-appointment tutor at Champlain College. Private tutor for dozens of clients over the past 8 years.

Publications
CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry

Education/Credentials
Bachelor of Science, Major Mathematics and Major Economics, McGill University, 2014. Diploma of Collegiate Studies; Pure and Applied Science, Champlain College Saint-Lambert, 2010.

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