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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

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

r²=a²+b²

r²=(1)²+(2)²

r²=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

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Comment | Dear Prof Azeem Thanks. Prashant |

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

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.