- #1

- 117

- 0

## Main Question or Discussion Point

Mathworld seems to be the only place which explicitly defines complex exponentiation. I'll be even more explicit than Wolfram and write out the full formula (I'm guessing there's no shorter way for the general case):

(a+b

(a

(a

My question is this:

Wikipedia says that the result for complex exponentiation is multivalued when the exponent is irrational - in other words, there's more than one possible solution. How would I change the above formula to produce the other solution/s?

If anyone could give the other answers to say (2.2+3.3i)^(4.4+5.5i) to clarify the answer to the above, then that would be a great example too. Or are there an infinite number of possible answers?

Finally, if complex exponentiation is supposed to give multiple answers when the exponent is irrational, then why do certain calculator applets on the internet that deal with complex arithmetic only give one answer?

(a+b

**)^(c+d***i***) =***i*(a

^{2}+b^{2})^(c/2) **e*^(-d*atan2(b,a)) * cos(c*atan2(b,a)+0.5*d*ln(a^{2}+b^{2}) )**+**(a

^{2}+b^{2})^(c/2) **e*^(-d*atan2(b,a)) * sin (c*atan2(b,a)+0.5*d*ln(a^{2}+b^{2}) )*i*My question is this:

Wikipedia says that the result for complex exponentiation is multivalued when the exponent is irrational - in other words, there's more than one possible solution. How would I change the above formula to produce the other solution/s?

If anyone could give the other answers to say (2.2+3.3i)^(4.4+5.5i) to clarify the answer to the above, then that would be a great example too. Or are there an infinite number of possible answers?

Finally, if complex exponentiation is supposed to give multiple answers when the exponent is irrational, then why do certain calculator applets on the internet that deal with complex arithmetic only give one answer?

Last edited: