Mathematical Analysis I (Advanced Calculus I)  [1ΚΠ01 - First Semester]

Sets. Mappings. Real numbers. Axiomatic foundations of the real numbers. Rational numbers. Intervals. Distance. Neighborhoods. Classification of points of R. Open and closed sets. Sequence of real numbers. Limit of a sequence. Operations with limits. The Cauchy criterion. Monotonic sequences. Contraction sequence. Recurrence sequence. Difference equation. Numerical series. Basic tests for convergence of numerical series.

Limit of a function. Operation with limits. Continuity of a function at a point. Operations with continuous functions. The derivative of a function. Basic theorems. Leibniz’s rule. Derivative of composite function. Many-valued functions. Inverse functions. Derivative of inverse functions. Inverse trigonometric functions. Hyperbolic functions. Inverse hyperbolic functions. Derivative of implicit functions. Change of variables in differential expressions. Differential of a function. Derivatives and differentials of hither order. Taylor’s formula with Peano’s form of the remainder. Taylor’s formula with Lagrange’s form of the remainder. Taylor’s series. Power series.

Indefinite integral. Basic methods of computing indefinite integral. The Riemman integral. Properties of the definite integral. The fundamental theorem of calculus. Further properties of integrals. Applications of the definite integral. Improper integral. Relationship between improper integrals and series. Basic tests for convergence of improper integrals.

Ordinary differential equations. Variable separate. Linear differential equations of first order. Linear differential equations of second order with constant coefficients. Euler- type equations.