Largest Region Where g(z) = 1/[Log(z)-(i*pi)/2] is Analytic

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In summary, the largest region in which g(z) = 1/[Log(z)-(i*pi)/2] is analytic is everywhere except for the positive imaginary axis when the argument of z is chosen to be between pi/2 and 5pi/2. This can be determined by considering the branch point and branch cut of Log(z), where the branch cut can be chosen arbitrarily from 0 to infinity. Both attempts at finding a solution involve considering the domain of g(z) and the properties of Log(z).
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Homework Statement



Describe the largest region in which g(z) = 1/[Log(z)-(i*pi)/2] is analytic?


The Attempt at a Solution



Analytic is where the cauchy-euler equations hold, so I tried to take the partials to and set them equal so I could define a domain. I am not sure if the partials are correct.

I also tried to look at it from the following point of view. It would be analytic where the domain of g(z) is valid. My line of thinking is Log(z) is the principal log and it has a branch cut at pi. Thus pi is not in the domain. Also 1/z where z≠0 so Log(z)-(i*pi)/2 ≠ 0.
Log(z) ≠ (i*pi)/2. I think Log(i) = (i*pi)/2. If so then the parts not in the domain are pi and i.

Does either attempt sound like a possible solution?
 
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  • #2
branch point of Log[z] is 0, the branch cut can be arbitrarily chosen to go from 0 to infinity, so if you choose arg(z) in (pi/2,5pi/2) then g(x) is analytic everywhere except positive imag axis
 

FAQ: Largest Region Where g(z) = 1/[Log(z)-(i*pi)/2] is Analytic

1. What is the definition of an analytic function?

An analytic function is a mathematical function that is infinitely differentiable on a certain domain and can be represented by a power series. It is also known as a complex differentiable function, meaning it can be differentiated with respect to complex numbers.

2. What is the significance of g(z) = 1/[Log(z)-(i*pi)/2] being analytic?

The fact that g(z) is analytic means that it has well-defined derivatives and can be approximated by a power series on its domain. This allows for the use of complex analysis techniques to study the behavior of g(z) and make predictions about its properties.

3. What is the largest region where g(z) = 1/[Log(z)-(i*pi)/2] is analytic?

The largest region where g(z) is analytic is the entire complex plane, with the exception of the points z = 0 and z = e^(i*pi/2). This is because the function is undefined at these points due to the singularity at Log(z) = (i*pi)/2.

4. How is the concept of analyticity related to the Cauchy-Riemann equations?

The Cauchy-Riemann equations are a set of necessary and sufficient conditions for a complex function to be analytic. These equations relate the partial derivatives of a complex function to its conjugate, and they must hold true for a function to be analytic.

5. Can a function be analytic at a single point?

No, a function must be analytic on a region in order for it to be considered analytic. This means that it must be infinitely differentiable on a non-empty open set in the complex plane. However, a function can be said to be holomorphic at a single point, which means it is complex differentiable at that point.

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