Syllabus
* Complex functions
Power series and its convergence radius, derivative and integral of power series.
Holomorphic functions, Cauchy-Riemann conditions, primitive functions, curve integral, Cauchy theorem,, Cauchy formula, Liouville theorem, power expansions of holomorphic functions, uniqueness theorem. Laurent series, residues and their application to integrals of real functions. Gamma function on complex numbers.
* Laplace and Fourier transforms
Basic properties and relations, transforms of elementary functions. Inverse Laplace and Fourier transforms. Application to solution of differential equations.
* Calculus of variations.
Extremal values of L(y)=Integral( f(x,y(x),y'(x)) , dx) and Euler equations, isoperimetrical problems.
Annotation
The fourth part of a four-semester course in calculus for bachelor's program Financial Mathematics.