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

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1. Feb 16, 2015

### monnapomona

1. The problem statement, all variables and given/known data
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)

2. Relevant equations

3. 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)?

2. Feb 16, 2015

### RUber

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

3. Feb 16, 2015

### monnapomona

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))

4. Feb 16, 2015

### RUber

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