What are Analytic Functions?

Welcome back MechanicaLEi, did you know that all elementary functions like polynomials, exponential functions, trigonometric functions, absolute value functions etc. are analytic functions? This makes us wonder, what are analytic functions? Before we jump in check out the previous part of this series to learn about what complex variables are? Now, A function f(z) is said to be analytic in a region R of the complex plane if f(z) has a derivative at each point of R and if f(z) is single valued. Also, A function f(z) is said to be analytic at a point z, if z is an interior point of some region where f(z) is analytic, that is, the concept of analytic function at a point implies that the function is analytic in some circle with its center at this point. Let f(x, y) = u(x, y) + i into v(x, y) be a complex function. Since x = (z + z bar)/2 and y = (z - z bar)/2i, substituting for x and y gives f(z, z bar) = u(x, y) + i into v(x, y). A necessary condition for f(z, z bar) to be analytic is dou f upon dou z equals zero. Mark it as 1. Therefore, a necessary condition for f = u + iv to be analytic is that f depends only on z. In terms of the of the real and imaginary parts u, v of f, condition (1) is equivalent to dou u upon dou x equals dou v upon dou y, mark this as 2 and dou u upon dou y equals minus dou v upon dou x, mark this as 3. Equations (2 and 3) are known as the Cauchy-Riemann equations. They are a necessary condition for f = u + iv to be analytic. The necessary and sufficient conditions however are two in numbers: first, the four partial derivatives of real and imaginary parts should satisfy the Cauchy-Riemann equations and second, the four partial derivatives of its real and imaginary parts are continuous. Hence, we first saw what analytic functions are and then went on to see the necessary and sufficient conditions for it?

In the next episode of MechanicaLEi find out what Milne-Thompson methods are?

Attributions:

Doh De Oh by Kevin MacLeod is licensed under a Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/)
Source: http://incompetech.com/music/royalty-free/index.html?isrc=USUAN1100255
Artist: http://incompetech.com/

Subtle Library by Fabian Measures (http://freemusicarchive.org/music/Fabian_Measures/) is licensed under a Creative Commons Attribution licence ( https://creativecommons.org/licenses/by/4.0/)
Source: http://freemusicarchive.org/music/Fabian_Measures/Emotion_Music/Subtle_Library