FURTHER COMPLEX METHODS - RECOMMENDED TEXT BOOKS :

*Complex Variables: Introduction and Applications* - M.J. Ablowitz and A.S. Fokas

*Modern Analysis* - E.T. Whittaker and G.N. Watson.

*Introduction to Complex Analysis* - H.A. Priestley (Clarendon)

*An Introduction to Functions of a Complex Variable* - E.T. Copson (Oxford).

*Methods of Mathematical Physics* - J. Mathews and R.L. Walker.

*Mathematical Physics* - H. and B.J. Jeffreys.

*Functions of a Complex Variable* - G Carrier, M. Krook and C. Pearson

*Handbook of Mathematical Functions* - M. Abramowitz and I. A. Stegun.

FCM Overview:

Revision of relevant material from Part IB Complex Methods.

The Riemann sphere. Picard's theorems.

Algebraic determination of an analytic function from its real part.

Functions defined by integrals: existence and analyticity; the Gaussian integral as an example.

Hand-out: Functions defined by integrals (an example).

Analytic continuation: the identity theorem, definition of analytic continuation, examples.

Hand-out: Analytic continuation by contour deformation (an example).

Branch points, branch cuts and multivalued functions, examples.

The branches of (1 - z^{2}) \ 12. Integration using a branch cut: example.

Hand-out: arcsin as an integral.

Cauchy principal value. Hilbert transforms.

Hand-out: Cauchy Principal Value (two examples).

The response function. Causality. The harmonic response plot.

Hand-out: Harmonic response plot (example).

Revision of the Laplace transform and application to PDEs.

Hand-out: Waves on a string.

Revision of the Fourier transform and application to PDEs.

Hand-out: The causal Green's function for the wave equation.

Solution of differential equations by integral representation.

Hand-out: The Airy equation

Solution of differential equations by integral transform continued.

Hand-out: The Hermite equation.

The Gamma function: Euler integral, product formulae, recurrence formula.

Hand-out: Product formulae.

Gamma function: reflection formula;

Hankel representation; Gamma on the real axis.

Hand-out: Hankel representation

Gamma function: uniqueness (Wielandt's theorem).

Beta function: Euler integral, Beta in terms of Gammas (two proofs).

Hand-out: Wielandt's theorem

Beta function: Pochhammer representation.

Zeta function; prime number formula, integral representations.

Hand-out: Prime number formula for the zeta function.

Zeta function: the functional equation. Riemann hypothesis.

Hand-out: The functional equation for the zeta function

2nd order ODEs: classification of singular points, including the point at infinity. Series solutions.

Hand-out: Solution of ODEs by series

The nature of solutions near an isolated singular point.

Hand-out: The log solution from the Wronskian

Equations with exactly three regular singular points (Papperitz) .

Hand-out: The Papperitz equation

The Riemann P function

The hypergeometric function and the hypergeometric equation

Hand-out: Legendre's equation

The hypergeometric equation and the hypergeometric equation

Hand-out: The hypergeometric equation (second solution near z = 0 and solutions near z = infinity.)

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