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Calculus/Compute exponentiation of two complex numbers.

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Question
Dear Prof Paul

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
Let z1 = x1 + i y1

Let z2 = x2 + i y2

If  z =  z1^z2, then

ln(z) = z2 ln(z1)

Now ln(z1) =  ln|z1| + i arg z1

and you can multiply this out with  z2,

then use the fact that

exp(z) = exp(x)( cos y + i sin y)

to recover the answer.

Any good complex variables book will have this.

Calculus

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Paul Klarreich

Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

Experience

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

Education/Credentials
(See above.)

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