Finding Analyticity Region of a Function - Matt

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In summary, the speaker is confused about how to find the analyticity region of a function. They mention the definition of an analytic function and give examples of functions that are analytic on certain regions but not on others. They ask if there is a way to compute or calculate the region instead of doing it mentally. They also bring up the concept of analytic continuation.
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Hi, I have some questions regarding how to find the analytisity region of a funtion.
I'm a little confuse after I studied the definition of analytic function: which it saids
[if a function f is differentiable at every z in A, then f is analytic on A]

eg. Log z is analytic on the entire complex plane EXCEPT the -ve real axis.
Which make sense to me since Log z is undefind when x<=0 & y=0 , for z=x+iy

Log z^2 is analytic on the entire complex plane again EXCEPT z=0, and exclude the
Imaginary axis. Is that right?

I'm wondering if there's a way to actually compute/calculate the region instead of doing it in the head?
Since Log z & Log z^2 is kinda basic, it'll be hard to do if it is comething like Log (1+2/z)

Thanks in advance

Matt
 
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log(-1) CAN be defined since e^(i*pi)=-1. The upper case 'L' in Log is not is not just ornamental. It's a branch of log that's undefined for negative reals precisely so it can be uniquely defined for all other complex numbers. You can't compute why a function like that is undefined. Where it's undefined is a matter of convention. Look up "analytic continuation".
 

1. What is the "analyticity region" of a function?

The analyticity region of a function refers to the set of points in the complex plane where the function is analytical, meaning it can be represented by a convergent power series. This region is also known as the domain of analyticity.

2. How do you determine the analyticity region of a function?

To determine the analyticity region of a function, you can use the Cauchy-Riemann equations, which state that a function is analytic at a point if it satisfies a set of partial differential equations. Alternatively, you can also use theorems such as the Cauchy Integral Theorem or the Cauchy Integral Formula to determine the analyticity region.

3. Can a function have more than one analyticity region?

Yes, a function can have multiple analyticity regions. This is because the analyticity region of a function may be different for different parts of the complex plane. For example, a function may be analytic in the upper half-plane but not in the lower half-plane.

4. How does the analyticity region affect the behavior of a function?

The analyticity region of a function can greatly affect its behavior. Inside the analyticity region, the function is smooth and can be represented by a power series, making it easier to analyze and manipulate. However, outside the analyticity region, the function may exhibit discontinuities, singularities, or other complex behavior.

5. Why is finding the analyticity region important in mathematics?

Finding the analyticity region of a function is important because it allows us to understand the behavior of the function and its properties. It also helps us in solving problems, such as finding the maximum or minimum values of a function or evaluating complex integrals. Additionally, the analyticity region is closely related to concepts such as holomorphy, which have important applications in physics and engineering.

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