Cauchy Riemann Equations and Differentiability | Analytic VS Holomorphic | Complex Analysis #2

The definition of the Cauchy Riemann Equations with example and how these equations are connected to Complex Differentiability, Analytic/Holomorphic functions and Entire functions. Examples for each concept are included.

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IMPORTANT LINKS
Recap about the Cauchy Riemann Equations: https://goo.gl/QVM8pG
Cauchy Riemann Equations in polar form and cartesian form: https://goo.gl/Qxp3SS

CONCEPTS FROM THE VIDEO
► Cauchy Riemann Equations (CRE):
If f(x) = u(x,y)+iv(x,y) is a complex function then the two partial differential equations du/dx = dv/dy and du/dy= -dv/dx are called the Cauchy Riemann Equations.

► If f(x) = u(x,y)+iv(x,y) is differentiable at z_0, then the Cauchy
Riemann Equations holds at z_0.

► Complex Differentiability
A function f(z) = u(x,y)+iv(x,y) is differentiable in a region R if and only if the following conditions are fulfilled in R:
1) du/dx, dv/dy, du/dy, dv/dx are continous
2) du/dx, dv/dy, du/dy, dv/dx satisfies the Cauchy Riemann Equations
The derivative is defined as f'(z) =du/dx + i*dv/dx=du/dy - i*dv/dy

► Analytic Functions (Holomorphic Functions)
A function f(z) is said to be analytic in a region R if the function is differentiable in a neighborhood of every point in R.

Note that the terms analytic function and holomorphic function are used interchangeably in complex analysis.

► Corollary:
If f(z) is analytic in a region R, then:
1) derivatives of all orders exist in R.
2) f(z) can be represented with a power series.

► Discussion: Analytic Versus Holomorphic
There are two sides here, the first one have the standpoint that
a holomorphic function is not defined the same as a analytic
function. The second standpoint is that these two words are
simply synonyms for each other (like in graph theory people
use vertex or node interchangeably).

Different webpages have taken different sides in this:
Wolfram for examples states on their page about Holomorphic
functions that "A synonym for analytic function,..." (Weisstein,
Eric W. "Holomorphic Function." From MathWorld--A Wolfram
Web Resource.
http://mathworld.wolfram.com/Holomorp...)

While Wikipedia states that: "Though the term analytic function
is often used interchangeably with "holomorphic function", the
word "analytic" is defined in a broader sense" (https://en.wikipedia.org/wiki/Holomor...)

► Analytic Versus Holomorphic Additional Reading:
https://math.stackexchange.com/questi...

http://mathworld.wolfram.com/Analytic...

http://mathworld.wolfram.com/Holomorp...

https://math.stackexchange.com/questi...

https://en.wikipedia.org/wiki/Holomor...

https://en.wikipedia.org/wiki/Analyti...

► Entire Functions
A function f(z) is an entire function if and only if the function is analytic on the complete complex plane.

► Analytic Continuation
It provides a way of extending the domain over which a complex function is defined. Let f_1 and f_2 be analytic functions which are defined on the domains d_1 and d_2, if f_1 = f_2 is true in the intersection of the domains then f_2 is called an analytic continuation of f_1 to d_2 and vice versa. This analytic continuation is unique if it exists.

TIMESTAMPS
Intro: 00:00 - 00:24
The Cauchy Riemann Theorem: 00:24 - 00:57
Complex Differentiability Theorem: 00:57 - 01:40
Example about Complex Differentiability: 01:40 - 04:27
Analytic continuation: 04:27 - 05:29
Analytic Functions part 1: 05:29 - 05:56
Discussion about Analytic versus Holomorphic: 05:56 - 06:59
Analytic Functions part 2: 06:59 - 07:27
Entire Functions: 07:27 - 07:40
Example Analytic and Entire functions: 07:40 - 09:52
Outro: 09:52 - 10:11

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