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This book comprises of the topics on Differential Equation, Laplace Transform, Uniform Convergence of Sequence and Series of Functions, Power Series and Fourier Series for the students.

This book provides a thorough exposition of the main concepts and results related to various types of convergence of measures arising in measure theory, probability theory, functional analysis, partial differential equations, mathematical physics, and other theoretical and applied fields. Particular attention is given to weak convergence of measures. The principal material is oriented toward a broad circle of readers dealing with convergence in distribution of random variables and weak convergence of measures. The book contains the necessary background from measure theory and functional analysis. Large complementary sections aimed at researchers present the most important recent achievements. More than 100 exercises (ranging from easy introductory exercises to rather difficult problems for experienced readers) are given with hints, solutions, or references. Historic and bibliographic comments are included. The target readership includes mathematicians and physicists whose research is related to probability theory, mathematical statistics, functional analysis, and mathematical physics.

Let $f$ be a periodic measurable function and $(n_k)$ an increasing sequence of positive integers. The authors study conditions under which the series $\sum_{k=1}^\infty c_k f(n_kx)$ converges in mean and for almost every $x$. There is a wide classical literature on this problem going back to the 30's, but the results for general $f$ are much less complete than in the trigonometric case $f(x)=\sin x$. As it turns out, the convergence properties of $\sum_{k=1}^\infty c_k f(n_kx)$ in the general case are determined by a delicate interplay between the coefficient sequence $(c_k)$, the analytic properties of $f$ and the growth speed and number-theoretic properties of $(n_k)$. In this paper the authors give a general study of this convergence problem, prove several new results and improve a number of old results in the field. They also study the case when the $n_k$ are random and investigate the discrepancy the sequence $\{n_kx\}$ mod 1.