# Where are these functions analytic?

In summary, an analytic function is a complex-valued function that can be represented by a convergent power series in some region of the complex plane. To determine if a function is analytic, you can check if it satisfies the Cauchy-Riemann equations or if it can be represented by a convergent power series. The region of analyticity for a function is the set of points in the complex plane where the function is analytic, which can be determined by analyzing the Cauchy-Riemann equations or by finding the radius of convergence for the power series. Not all functions are analytic, as they must satisfy certain conditions. A function can also be analytic at some points and not others, meaning it is only analytic in certain regions of
Greetings all, I have 2 functions:

$$\frac{1}{Rez} + \frac{1}{Imz}(z^{2}- \overline{z}^{2})$$

and

$$\frac{1}{Rez} + \frac{1}{i Imz}(z^{2}- \overline{z}^{2})$$

I have to find where they are analytic, how do I start this?

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A good place to start is to look up the definition of analytic

You may also want to look up "Cauchy-Riemann equations". They provide a simple test for the analyticity of a function.

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## 1. What does it mean for a function to be analytic?

An analytic function is a complex-valued function that can be represented by a convergent power series in some region of the complex plane. This means that the function is smooth and has a continuous derivative of all orders in this region.

## 2. How can I determine if a function is analytic?

To determine if a function is analytic, you can check if it satisfies the Cauchy-Riemann equations, which are necessary and sufficient conditions for a function to be analytic. You can also check if the function can be represented by a convergent power series.

## 3. What is the region of analyticity for a function?

The region of analyticity for a function is the set of points in the complex plane where the function is analytic. This region can be determined by analyzing the Cauchy-Riemann equations or by finding the radius of convergence for the power series representation of the function.

## 4. Are all functions analytic?

No, not all functions are analytic. For a function to be analytic, it must satisfy certain conditions such as being smooth and having a continuous derivative of all orders. Many common functions, such as absolute value and step functions, are not analytic.

## 5. Can a function be analytic at some points and not others?

Yes, a function can be analytic at some points and not others. This means that the function is only analytic in certain regions of the complex plane and not others. An example of this is the function 1/z, which is analytic everywhere except at z=0.

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