Topics in an Analysis Course
To learn more about a topic listed below, click the topic name to go to the
corresponding MathWorld classroom page.
||(1) In higher mathematics, analysis is the systematic study of real- and complex-valued continuous functions. (2) In the mathematical theory of logic, analysis refers to second-order arithmetic.
||A Bernoulli number is one in a sequence of signed rational numbers that can be defined using a certain simple generating function. Bernoulli numbers are very important in number theory and analysis, and commonly arise in series expansions of trigonometric functions.
|Calculus of Variations
||The calculus of variations is a generalization of the usual calculus that seeks to find the path, curve, surface, etc., for which a given function has a stationary value (which, in physical problems, is usually a minimum or maximum).
||A Cantor set is a particular example of an uncountable set of measure zero constructed from a unit interval by recursively removing the middle third of subintervals.
||Convolution is the integral transform that expresses the amount of overlap of one function g as it is shifted over another function f.
||The delta function, also called the Dirac delta function, is a generalized function that has the property that its convolution with any function f equals the value of f at zero.
||A Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines.
||The gamma function is an extension of the factorial to real and complex arguments.
||Lebesgue measure is an extension of the classical notions of length and area to more complicated sets.
||A measure is a function that quantifies the size of a subset of a Euclidean space. Measures are used for integration and are important in differential equations and probability theory.
||A Banach space is a vector space that has a complete norm. Banach spaces are important in the study of infinite-dimensional vector spaces.
||Functional analysis is a branch of mathematics concerned with infinite-dimensional vector spaces and mappings between them.
||A Hilbert space is a vector space that has a complete inner product. Hilbert spaces are important in the study of infinite-dimensional vector spaces.