Analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, sequences, series, and analytic functions. 

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, applied mathematics; as well as in physics. Complex analysis is particularly concerned with the analytic functions of complex variables (or, more generally, meromorphic functions). Because the separate real and imaginary parts of any analytic function must satisfy Laplace's equation, complex analysis is widely applicable to two-dimensional problems in physics.


Stephen Brady -  Numerical Linear Algebra
Tom DeLillo - Numerical conformal mapping, fluid mechanics, computational methods for inverse problems
Robert Fraser - Harmonic Analysis
Buma Fridman - Holomorphic functions of several complex variables. Classification of complex manifolds.
Christopher Green - Special function theory, conformal mapping, fluid mechanics
Lop-Hing Ho - Several Complex Variables