- #1
Peter VDD
- 6
- 0
What does it mean "an analytic function", "analyticity" of a function?
I know we have seen it but i cannot find it back.
I know we have seen it but i cannot find it back.
Galileo said:Satisfaction of the Cauchy-Riemann relations at a point may not guarantee the existence of f' at that point. If you add the assumption that all first order partial derivatives ([itex]u_x,u_y,v_x,v_y[/itex]) are continuous at a point, then that's sufficient to guarantee the analyticy of f at that point.
An analytic function is a mathematical function that is defined and continuous on a complex domain. It is also known as a holomorphic function and satisfies the Cauchy-Riemann equations, meaning that it is differentiable at every point in its domain.
Analytic functions are different from regular functions in that they are defined on a complex domain and satisfy the Cauchy-Riemann equations. Regular functions do not necessarily have these properties and can be defined on a real domain.
Examples of analytic functions include polynomials, trigonometric functions, and exponential functions. The complex logarithm and complex exponential functions are also analytic functions.
Analytic functions have many applications in mathematics and science, including in complex analysis, differential equations, and physics. They are also used in engineering, economics, and finance for modeling and solving problems.
To determine if a function is analytic, you can check if it satisfies the Cauchy-Riemann equations. You can also use the Cauchy integral theorem to test for analyticity. Additionally, many common functions are known to be analytic, so you can also check if your function falls into one of these categories.