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

In summary, the largest set D on which f(z) is analytic is D = C\ { +/- 1-2i } υ {-y+1} and its derivative is 1/(z^2+2z+5) * (1 - (-iz)/(z^2+2z+5)).
  • #1
monnapomona
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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)?
 
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  • #2
For the log, the restriction is on the real part.
For your derivative, it seems like you lost the contribution from the denominator.
 
  • #3
RUber said:
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))
 
  • #4
This would be either the product rule or the quotient rule.
##[\frac{g(z)}{f(z)}]'= \frac{fg'-gf'}{[f(z)]^2}##
 

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

1. What is complex analysis?

Complex analysis is a branch of mathematics that deals with complex numbers and functions of complex variables. It explores the properties of these numbers and functions and how they behave under mathematical operations.

2. What is an analytic function?

An analytic function is a complex-valued function that can be represented as a power series in a certain region of the complex plane. It is a function that is smooth and has a derivative at every point in its domain.

3. What is the largest set where a function is analytic?

The largest set where a function is analytic is known as its domain of analyticity. This is the set of points in the complex plane where the function is defined and has a derivative at every point. In general, the domain of analyticity can be a connected open set.

4. How can we determine the domain of analyticity for a function?

In order to determine the domain of analyticity for a function, we need to check for the existence of a derivative at every point in its domain. This can be done by using the Cauchy-Riemann equations, which state that a function is analytic if and only if its partial derivatives satisfy a certain set of conditions.

5. Why is the largest set where a function is analytic important?

The largest set where a function is analytic is important because it tells us where the function is well-behaved and can be used to make meaningful mathematical statements. It also helps us determine the behavior of the function near its boundaries, which can have important implications in various fields of mathematics and science.

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