Analysis 3 Proofs and Definitions Weeks 9 and 10

Basic notation and results in C (of complex numbers):

Convergence in complex numbers:

Open set in complex numbers:

Sequentially compact sets in complex numbers:

How is real differentiation is different to complex differentiation (key difference is you can divide by Z instead of linear maps)

Cauchy-Riemann equations

Trick to check cauchy-riemann equations using conjugate of Z ie derivative of f with respect to conjugate of z must be zero to be complex differentiable

Holomorphic, analytic and entire functions

Power series and convergence of power series

Ratio test

Root test

Radius of convergence of power series

Radius of convergence using ratio test

Differentiating term by term of power series (proof is too complicated)

Derivatives for fn(0) basically just factorials in this case

Power series converges for anything less than the radius of convergence

Exponential function

Trigonometric functions in terms of exponential power series

Argument and logarithmic functions for complex numbers

complex integral:

properties of complex integration

Complex integrals along curves (contour integral)

Contour integrals of reverse parameterisation and in the same direction

Length of a curve using f(z) = 1

Conjugate of integral is not equal to integral of conjugate

Fundamental theorem of calculus for complex line integrals

Connected and simply connected complex sets

Cauchy’s theorem in complex integration

Deformation of contours parameterisation

Solving the fundamental contour integral

Interior and exterior region of a complex curve

Theorem of analytic functions using deformation of curves