Real Analysis
Real analysis (traditionally the theory of functions of a real variable ) is a branch of mathematical analysis dealing with real numbers and real functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including the convergence and limitations of sequences of real numbers, the calculus of real numbers and continuity, smoothness, and related properties of functions with real values.

Complex Analysis
Complex analysis, traditionally known as the theory of functions of the complex variable, is a branch of mathematical analysis that investigates functions from complex numbers. It is useful in many areas of mathematics, including algebraic geometry, number theory, and applied mathematics ; and in physics, including fluid dynamics, thermodynamics, engineering, electrical engineering, and, in particular, quantum field theory.

Complex analysis is especially concerned with analytic functions of complex variables (or, more generally, with meromorphic functions ). Since the individual real and imaginary parts of any analytic function must satisfy the Laplace equation, complex analysis is widely applicable to two-dimensional problems in physics.

Functional Analysis
Functional analysis is a branch of mathematical analysis whose core is the study of vector spaces endowed with some structure related to constraints (e.g. inner product, norm, topology, etc.) and linear operators acting on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of function spaces and the formulation of properties of function transformations, such as the Fourier transform, as transformations defining continuous, unitary, etc. D. operators between function spaces. This viewpoint has proved particularly useful for the study of differential and integral equations.

Differential Equations.
A differential equation is a mathematical equation for an unknown function of one or more variables that relates the values of the function itself and its derivatives of various orders. Differential equations play an important role in engineering, physics, economics, biology, and other disciplines.

Differential equations arise in many areas of science and engineering, particularly when a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as time changes. Newton’s laws allow (taking into account the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically in the form of a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called the equation of motion ) can be solved explicitly.

Measurement theory
A measure on a set is a systematic way of assigning a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume.

Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulation ) for problems of mathematical analysis (as opposed to discrete mathematics ).

Modern numerical analysis doesn’t look for exact answers because exact answers are often impossible to get in practice. Instead, most numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable error bounds.

Numerical analysis naturally finds applications in all areas of engineering and the physical sciences, but in the 21st century, the life sciences and even the arts have embraced elements of scientific computation. Ordinary differential equations appear in celestial mechanics (planets, stars, and galaxies); numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are necessary for modeling living cells in medicine and biology.