What is Analytic functions?
A complex function is said to be analytic on a region R if it is complex differentiable at every point in. The terms holomorphic function, differentiable function, and complex differentiable function are sometimes used interchangeably with "analytic function".
The Cauchy Riemann equations for a pair of given real-valued functions in two variables say u (x, y) and v (x, y) are the following two equations:
In a typical way, the values ‘u’ and ‘v’ are taken as the real and the imaginary parts of a complex-valued function of a single complex variable respectively,
z=x+iy, g(x+iy) = u(x,y)+iv(x,y)
if we are given that the functions u and v are differentiable at real values at a point in an open subset of the set of complex numbers that is C which can be taken as functions that are from R2 to R. This will imply that the partial derivatives of u and v do exist and thus we can also approximate smaller variations of ‘g’ in linear form. Then we say that g=u+iv is differentiable at complex values at that particular point iff the Cauchy Riemann equations at that point are satisfied by the partial derivatives of u and v.
Cauchy Riemann Equations in Polar Coordinates:
Cauchy Integral Theorem
If is analytic in some simply connected region, then
for any closed contour completely contained in . Writing as
From Green's theorem,
But the Cauchy-Riemann equations require that
Taylor Series for Holomorphic Functions. In Real Analysis, the Taylor series of a given function f : R → R is given by:
We have examined some convergence issues and applications of Taylor series in MATH 2033/2043. We also learned that even if the function f is infinitely differentiable everywhere on R, its Taylor series may not converge to that function. In contrast, there is no such an issue in Complex Analysis: as long as the function f : C → C is holomorphic on an open ball B(z0), we can show the Taylor series of f.
The Laurent series is a representation of a complex function f(z) as a series. Unlike the Taylor series which expresses f(z) as a series of terms with non-negative powers of z, a Laurent series includes terms with negative powers. A consequence of this is that a Laurent series may be used in cases where a Taylor expansion is not possible.
To calculate the Laurent series we use the standard and modified geometric series which are:
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