Question

Define the principal branch of a* (for a Є C) by a* := eᶻ ᴸᵒᵍ⁽ᵃ⁾. Use the chain rule to compute the derivative of a. Where is this derivative valid?

Answer 1

The principal branch of a* is defined as a* = e^(z Log(a)). To find the derivative of a, we use the chain rule and find the derivative of Log(a) first. The derivative is valid for all complex numbers a and z.

Step-by-step explanation:

The principal branch of a* is defined as a* = e^(z Log(a)). To compute the derivative of a, we can use the chain rule. Let's start by finding the derivative of Log(a).

d/dz(Log(a)) = 1/a

Next, we can find the derivative of e^(z Log(a)). Using the chain rule, the derivative is:

d/dz(e^(z Log(a))) = e^(z Log(a)) * Log(a)

This derivative is valid for all complex numbers a and z.