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Cauchy Riemann Equations and Differentiability Analytic VS Holomorphic Complex Analysis 2
>> YOUR LINK HERE: ___ http://youtube.com/watch?v=41pu051ZJAo
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. • LINK TO COMPLEX ANALYSIS PLAYLIST • • The Complete Guide to Complex Analysi... • LINK TO CANVAS • I will start to include my canvas from this video and forward: https://drive.google.com/file/d/0B0vT... • SUPPORT • Consider subscribing and liking if you enjoyed this video or if it helped you understand the subject. It really helps me a lot. • 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 • SOCIAL • ► Follow me on Youtube: http://bit.ly/1NQhPJ9 • ► Follow me on Twitter: / the_mathcoach • HASHTAGS • #TheMathCoach #ComplexAnalysis
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