Syllabus
* Multiple integrals.
Basic properties, Fubini theorem, substitutions, polar and spherical coordinates, volumes.
* Measure theory. basic properties of measure, construction of measure from outer measure, measurable functions, integral based on measure, Jordan and Lebesgue measures.
* Function sequences and series.
Pointwise and uniform convergence (Weierstrass test), commutation of convergence and limits, derivatives and integrals. Power series and their convergence radius, derivatives and integrals.
* Integrals with parameters.
Commutation of integral with limits, series and derivatives, Gamma and Beta functions, application to more complicated integrals.
* Fourier series.
Trigonometric series, Fourier coeficients, Parseval equation, convergence of
Fourier series, application to series of numbers.
Annotation
The third part of a four-semester course in calculus for bachelor's program Financial Mathematics.