Imaginary and complex exponents ```Name: lord byron Status: N/A Age: N/A Location: N/A Country: N/A Date: Around 1995 ``` Question: How do you raise a number to an imaginary power? How about raising a number to a complex power? Replies: Actually, the way to understand how this is done is to represent raising a number to a power in a different way: ``` x^y = e^(y * log(x)) ``` That is, as long as we can take the logarithm of the number, and as long as we know what e to any power is (complex included) then we can raise that number to any complex power. Well, there is no problem taking the log of any positive real number - you cannot do it for zero though, and there are some problems with negative numbers - the logarithm is itself complex to start with! So we can at least do it for the positive reals. The exponen- tial of a complex number is also easy to evaluate: ` e^(a + i b) = e^a (cos(b) + i sin(b))` so for positive real x, with y = a + ib complex, the answer is: `x^(a + i b) = e^((a + i b) log(x))=x^a (cos(b log(x)) + i sin (b log(x)))` So in general, raising a number to an imaginary power leads to trigonometric functions of the logarithm of that number. asmith Click here to return to the Mathematics Archives

NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators, sponsored and operated by Argonne National Laboratory's Educational Programs, Andrew Skipor, Ph.D., Head of Educational Programs.

For assistance with NEWTON contact a System Operator (help@newton.dep.anl.gov), or at Argonne's Educational Programs