How do you take the derivative of x raised to a complex number? For example, let z = a + b*i and let y = x^z. How do you get dy/dx? I assume that you'd bring down the (a + b*i) to multiply the x, but what is the proper way to decrement the exponential term?

Question

How do you take the derivative of x raised to a complex number?  For example, let z = a + b*i and let y = x^z.  How do you get dy/dx?  I assume that you'd bring down the (a + b*i) to multiply the x, but what is the proper way to decrement the exponential term?

anonymous · Answer

Take ln!

anonymous · Answer

So long as b is zero, you multiply the exponent by a and subtract one from the exponent$$ax ^{a-1}$$I don't know about other cases of b

anonymous · Answer

Ln y     = ( a + bi) Ln x
->  y'/y  =  ( a + bi) / x
=> y'      =  ( a + bi) y/x

anonymous · Answer

Thanks, how did you get the 2nd step?

anonymous · Answer

Thanks @Chlorophyll !

anonymous · Answer

The way you write it, the power rule is true for all numbers. So, when solving your equation @schmidtsl just use the power rule I wrote above.

anonymous · Answer

Yes, I see that Chlorophyll's result leads to exactly that (which is what you'd expect).  Thanks.

anonymous · Answer

Oh, now I see what Chlorophyll meant.  In the 2nd step, you take the derivative with respect to x on both sides.  d(ln(y))/dx = (1/y) * dy/dx = y'/y and d(ln(x))/dx = 1/x.  Thank you!!

anonymous · Answer

@L.T.  @schmidtsl Sorry, I was away from my PC!