ANALYSIS I

Limits and convergence

Sequences and series in R and C. Sums, products and quotients. Absolute convergence; absolute convergence implies convergence. The Bolzano-Weierstrass theorem and applications (the General Principle of Convergence). Comparison and ratio tests, alternating series test. [6]

Continuity of real- and complex-valued functions deﬁned on subsets of R and C. The intermediate value theorem. A continuous function on a closed bounded interval is bounded and attains its bounds. [3]

Diﬀerentiability of functions from R to R. Derivative of sums and products. The chain rule. Derivative of the inverse function. Rolle’s theorem; the mean value theorem. One-dimensional version of the inverse function theorem. Taylor’s theorem from R to R; Lagrange’s form of the remainder. Complex diﬀerentiation. [5]

Complex power series and radius of convergence. Exponential, trigonometric and hyperbolic functions, and relations between them. *Direct proof of the diﬀerentiability of a power series within its circle of convergence*. [4]

Deﬁnition and basic properties of the Riemann integral. A non-integrable function. Integrability of monotonic functions. Integrability of piecewise-continuous functions. The fundamental theorem of calculus. Diﬀerentiation of indeﬁnite integrals. Integration by parts. The integral form of the remainder in Taylor’s theorem. Improper integrals. [6]

T.M. Apostol Calculus, vol 1. Wiley 1967-69(djvu)
† J.C. Burkill A First Course in Mathematical Analysis. Cambridge University Press 1978(pdf)
D.J.H.Garling A Course in Mathematical Analysis (Vol 1). Cambridge University Press 2013(pdf)
J.B. Reade Introduction to Mathematical Analysis. Oxford University Press
M. Spivak Calculus. Addison–Wesley/Benjamin–Cummings 2006(pdf)
David M. Bressoud A Radical Approach to Real Analysis. Mathematical Association of America Textbooks(pdf)