Complex Analysis: Largest set where f(z) is analytic

[monnapomona]

Homework Statement

Find the largest set D on which f(z) is analytic and find its derivative. (If a branch is not specified, use the principal branch.)

f(z) = Log(iz+1) / (z^2+2z+5)

Homework Equations

The Attempt at a Solution

Not sure how to even attempt this solutions but I wrote down that
iz+1 ∉ (-∞,0]. This is where I get confused! Not sure if I have to put z in x+iy form.

For the denominator, z^2+2z+5 ≠ 0 implies z = +/- 1-2i.

So my incomplete solution would be D = C\ { +/- 1-2i } υ { ?? } and the derivative is 1/(iz+1)?

 

[RUber]

For the log, the restriction is on the real part.
For your derivative, it seems like you lost the contribution from the denominator.

 

[monnapomona]

For the log, the restriction is on the real part.
For your derivative, it seems like you lost the contribution from the denominator.

Okay, so if it's just the real part, iz+1 = i(x+iy) + 1 = ix - y +1 so the restriction would just be -y+1, where y ≠ 1?

I'm unsure what to do for a derivative, in my class notes it states that [log z ]' = 1/z so would it include the whole f(z) function, ie. ((z^2 + 2z + 5) / (iz+1))

 

[RUber]

This would be either the product rule or the quotient rule.
$[\frac{g(z)}{f(z)}]'= \frac{fg'-gf'}{[f(z)]^2}$