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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 - 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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