# Finding analyticity of a complex function involving ln(iz)

• tixi
In summary, the conversation discusses finding the largest region in which the function f(z) is analytical and how to differentiate it. The function ln (principal value of ln) is analytical in a complex plane with a removed negative real axis, denoted by D. The function f(z) has a domain of analyticity obtained from D by a rotation induced by the argument iz. The chain rule, ##\tfrac{d}{dz}\bar{\ln}(h(z))=\tfrac{1}{h(z)}\cdot h^\prime(z)##, can be used to differentiate the function.
tixi
Homework Statement
The function ln (principal value of ln) is known to be analytical in complex plane with removed negative real axis. What is the largest region in which f(z)=ln(iz)-i pi/2 is analytical? Evaluate f′(z).
Relevant Equations
Principal value of the logarithm: ln(z) = ln(r) + iArg(z)
Chain rule for complex functions
Inverse functions differentiation rule
Hey everyone! I got stuck with one of my homework questions. I don't 100% understand the question, let alone how I should get started with the problem.

The picture shows the whole problem, but I think I managed doing the a and b parts, just got stuck with c. How do I find the largest region in which f(z) is analytical and how do I get started trying to differentiate it? Do the differentiation rules for the real ln translate to the complex one?

Start here: The function ln (principal value of ln) is known to be analytical in complex plane with removed negative real axis, let this region be denoted by ##D##.
The function ##f(z)=\bar{\ln} (iz)-i\tfrac{\pi}{2}## has a domain of analyticity obtained from ##D## by a rotation (this rotation is induced by the argument ##iz##). Think you can figure the rest of that out?
My complex is rusty, but if I recall correctly ##\tfrac{d}{dz}\bar{\ln }(h(z))=\tfrac{1}{h(z)}\cdot h^\prime (z)## by the chain rule. That should get you going!

Thank you so much! I never got a notification that my thread was answered, but this still helped! The exercise was explained eventually but it was very abstract and they just gave a very simple answer, so the motivation you provided was still helpful! Thanks

benorin

## 1. What is the definition of analyticity in complex analysis?

Analyticity in complex analysis refers to the property of a complex function to be differentiable at every point in its domain. This means that the function has a well-defined derivative at every point, allowing for the use of techniques such as power series and Cauchy's integral formula.

## 2. How do you determine if a complex function involving ln(iz) is analytic?

To determine if a complex function involving ln(iz) is analytic, we must first rewrite the function in terms of its real and imaginary parts. Then, we can use the Cauchy-Riemann equations to check if the partial derivatives of the real and imaginary parts exist and are continuous at a given point. If they are, then the function is analytic at that point.

## 3. Can a complex function involving ln(iz) be analytic at some points and not others?

Yes, it is possible for a complex function involving ln(iz) to be analytic at some points and not others. This is because the Cauchy-Riemann equations only guarantee analyticity at a specific point, and not throughout the entire domain of the function.

## 4. What is the relationship between analyticity and the existence of a complex logarithm?

Analyticity and the existence of a complex logarithm are closely related. A complex function involving ln(iz) is analytic if and only if it has a well-defined complex logarithm. This means that the function must have a single-valued logarithm throughout its domain.

## 5. Are there any special cases where a complex function involving ln(iz) is always analytic?

Yes, there are special cases where a complex function involving ln(iz) is always analytic. One example is when the function is a constant, as the partial derivatives of the real and imaginary parts will always be 0. Another example is when the function is a polynomial, as all polynomials are analytic in the complex plane.

Replies
3
Views
786
Replies
17
Views
2K
Replies
2
Views
1K
Replies
5
Views
1K
Replies
1
Views
1K
Replies
3
Views
1K
Replies
4
Views
1K
Replies
6
Views
1K
Replies
3
Views
1K
Replies
9
Views
2K