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.
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 - z2) \ 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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