Analytic Function of Complex Variable z=x+iy

In summary, an analytic function of a complex variable is a function that is differentiable at every point in its domain and must have a well-defined derivative. The concept of analytic functions is closely related to the complex plane, as it can be represented as a mapping from the complex plane to itself. The main properties of analytic functions include being differentiable at every point in the complex plane, having a well-defined derivative, and satisfying the Cauchy-Riemann equations. They are widely used in mathematics and science, particularly in fields such as complex analysis, differential equations, physics, and engineering. To determine if a function is analytic, one can check if it satisfies the Cauchy-Riemann equations or if it can be expressed as a
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Which of the following is an analytic function of the the complex variable z = x+iy in the domain lzl <2 ?

(3+x-iy)^7
[(1+x+iy)^4]*[(7-x-iy)^3]
[(1-2x-iy)^4]*[(3-x-iy)^3]
(x+iy-1)^1/2
 
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You need to set out your post in the format required for the homework forum. Also, you should give an attempt at calculating them.
 

1. What is an analytic function of a complex variable?

An analytic function of a complex variable is a function that is differentiable at every point in its domain. In other words, the function must have a well-defined derivative at every point in the complex plane.

2. How is the concept of analytic functions related to the complex plane?

The concept of analytic functions is closely related to the complex plane because it is defined as a function of a complex variable. This means that the function can be represented as a mapping from the complex plane to itself.

3. What are the main properties of analytic functions?

The main properties of analytic functions include being differentiable at every point in the complex plane, having a well-defined derivative, and satisfying the Cauchy-Riemann equations. These properties allow for the use of powerful tools such as the Cauchy integral theorem and the Cauchy integral formula.

4. How are analytic functions used in mathematics and science?

Analytic functions are used in a wide range of mathematical and scientific fields, including complex analysis, differential equations, physics, and engineering. They are particularly useful in solving problems involving complex numbers and in the study of functions with complex inputs and outputs.

5. How can one determine if a function is analytic or not?

To determine if a function is analytic, one can check if it satisfies the Cauchy-Riemann equations, which are necessary conditions for a function to be analytic. Additionally, if a function is expressible as a power series, then it is also analytic. Other methods include using the Cauchy-Riemann integral theorem or the Cauchy-Riemann integral formula.

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