First Edition, 2012 ISBN 978-81-323-4247-2 © All rights reserved. Published by: White Word Publications 4735/22 Prakashdeep Bldg, Ansari Road, Darya Ganj, Delhi - 110002 Email: [email protected] Table of Contents Chapter 1 - Additive Function Chapter 2 - Algebraic Function Chapter 3 - Analytic Function Chapter 4 - Completely Multiplicative Function and Concave Function Chapter 5 - Convex Function Chapter 6 - Differentiable Function Chapter 7 - Elementary Function and Entire Function Chapter 8 - Even and Odd Functions Chapter 9 - Harmonic Function Chapter 10 - Holomorphic Function Chapter 11 - Homogeneous Function Chapter 12 - Indicator Function Chapter 13 - Injective Function Chapter 14 - Measurable Function Chapter 15 - Meromorphic Function Chapter 16 - Multiplicative Function and Multivalued Function Chapter 17 - Periodic Function Chapter 18 - Pseudoconvex Function and Quasiconvex Function Chapter 19 - Rational Function Chapter 20 - Ring of Symmetric Functions Chapter 1 Additive Function In mathematics the term additive function has two different definitions, depending on the specific field of application. In algebra an additive function (or additive map) is a function that preserves the addition operation: f(x + y) = f(x) + f(y) for any two elements x and y in the domain. For example, any linear map is additive. When the domain is the real numbers, this is Cauchy's functional equation. In number theory, an additive function is an arithmetic function f(n) of the positive integer n such that whenever a and b are coprime, the function of the product is the sum of the functions: f(ab) = f(a) + f(b). Completely additive An additive function f(n) is said to be completely additive if f(ab) = f(a) + f(b) holds for all positive integers a and b, even when they are not co-prime. Totally additive is also used in this sense by analogy with totally multiplicative functions. If f is a completely additive function then f(1) = 0. Every completely additive function is additive, but not vice versa. Examples Example of arithmetic functions which are completely additive are:  The restriction of the logarithmic function to N.  The multiplicity of a prime factor p in n, that is the largest exponent m for which pm divides n.  a (n) - the sum of primes dividing n counting multiplicity, sometimes called 0 sopfr(n), the potency of n or the integer logarithm of n (sequence A001414 in OEIS). For example: a (4) = 2 + 2 = 4 0 a (20) = a (22 · 5) = 2 + 2+ 5 = 9 0 0 a (27) = 3 + 3 + 3 = 9 0 a (144) = a (24 · 32) = a (24) + a (32) = 8 + 6 = 14 0 0 0 0 a (2,000) = a (24 · 53) = a (24) + a (53) = 8 + 15 = 23 0 0 0 0 a (2,003) = 2003 0 a (54,032,858,972,279) = 1240658 0 a (54,032,858,972,302) = 1780417 0 a (20,802,650,704,327,415) = 1240681 0  The function Ω(n), defined as the total number of prime factors of n, counting multiple factors multiple times, sometimes called the "Big Omega function" (sequence A001222 in OEIS). For example; Ω(1) = 0, since 1 has no prime factors Ω(20) = Ω(2·2·5) = 3 Ω(4) = 2 Ω(27) = 3 Ω(144) = Ω(24 · 32) = Ω(24) + Ω(32) = 4 + 2 = 6 Ω(2,000) = Ω(24 · 53) = Ω(24) + Ω(53) = 4 + 3 = 7 Ω(2,001) = 3 Ω(2,002) = 4 Ω(2,003) = 1 Ω(54,032,858,972,279) = 3 Ω(54,032,858,972,302) = 6 Ω(20,802,650,704,327,415) = 7 Example of arithmetic functions which are additive but not completely additive are:  ω(n), defined as the total number of different prime factors of n (sequence A001221 in OEIS). For example: ω(4) = 1 ω(20) = ω(22·5) = 2 ω(27) = 1 ω(144) = ω(24 · 32) = ω(24) + ω(32) = 1 + 1 = 2 ω(2,000) = ω(24 · 53) = ω(24) + ω(53) = 1 + 1 = 2 ω(2,001) = 3 ω(2,002) = 4 ω(2,003) = 1 ω(54,032,858,972,279) = 3 ω(54,032,858,972,302) = 5 ω(20,802,650,704,327,415) = 5  a (n) - the sum of the distinct primes dividing n, sometimes called sopf(n) 1 (sequence A008472 in OEIS). For example: a (1) = 0 1 a (4) = 2 1 a (20) = 2 + 5 = 7 1 a (27) = 3 1 a (144) = a (24 · 32) = a (24) + a (32) = 2 + 3 = 5 1 1 1 1 a (2,000) = a (24 · 53) = a (24) + a (53) = 2 + 5 = 7 1 1 1 1 a (2,001) = 55 1 a (2,002) = 33 1 a (2,003) = 2003 1 a (54,032,858,972,279) = 1238665 1 a (54,032,858,972,302) = 1780410 1 a (20,802,650,704,327,415) = 1238677 1 Multiplicative functions From any additive function f(n) it is easy to create a related multiplicative function g(n) i.e. with the property that whenever a and b are coprime we have: g(ab) = g(a) × g(b). One such example is g(n) = 2f(n). Chapter 2 Algebraic Function In mathematics, an algebraic function is informally a function that satisfies a polynomial equation whose coefficients are themselves p olynomials. For example, an algebraic function in one variable x is a solution y for an equation where the coefficients a(x) are polynomial functions of x. A function which is not i algebraic is called a transcendental function. In more precise terms, an algebraic function may not be a function at all, at least not in the conventional sense. Consider for example the equation of a circle: This determines y, except only up to an overall sign: However, both branches are thought of as belonging to the "function" determined by the polynomial equation. Thus an algebraic function is most naturally considered as a multiple valued function. An algebraic function in n variables is similarly defined as a function y which solves a polynomial equation in n + 1 variables: It is normally assumed that p should be an irreducible polynomial. The existence of an algebraic function is then guaranteed by the implicit function theorem. Formally, an algebraic function in n variables over the field K is an element of the algebraic closure of the field of rational functions K(x ,...,x ). In order to understand 1 n algebraic functions as functions, it becomes necessary to introduce ideas relating to Riemann surfaces or more generally algebraic varieties, and sheaf theory. Algebraic functions in one variable Introduction and overview The informal definition of an algebraic function provides a number of clues about the properties of algebraic functions. To gain an intuitive understanding, it may be helpful to regard algebraic functions as functions which can be formed by the usual algebraic operations: addition, multiplication, division, and taking an nth root. Of course, this is something of an oversimplification; because of casus irreducibilis (and more generally the fundamental theorem of Galois theory), algebraic functions need not be expressible by radicals. First, note that any polynomial is an algebraic function, since polynomials are simply the solutions for y of the equation More generally, any rational function is algebraic, being the solution of Moreover, the nth root of any polynomial is an algebraic function, solving the equation Surprisingly, the inverse function of an algebraic function is an algebraic function. For supposing that y is a solution of for each value of x, then x is also a solution of this equation for each value of y. Indeed, interchanging the roles of x and y and gathering terms, Writing x as a function of y gives the inverse function, also an algebraic function. However, not every function has an inverse. For example, y = x2 fails the horizontal line test: it fails to be one-to-one. The inverse is the algebraic "function" . In this sense, algebraic functions are often not true functions at all, but instead are multiple valued functions. The role of complex numbers From an algebraic perspective, complex numbers enter quite naturally into the study of algebraic functions. First of all, by the fundamental theorem of algebra, the complex numbers are an algebraically closed field. Hence any polynomial relation p(y, x) = 0 is guaranteed to have at least one solution (and in general a number of solutions not exceeding the degree of p in x) for y at each point x, provided we allow y to assume complex as well as real values. Thus, problems to do with the domain of an algebraic function can safely be minimized. A graph of three branches of the algebraic function y, where y3 − xy + 1 = 0, over the domain 3/22/3 < x < 50. Furthermore, even if one is ultimately interested in real algebraic functions, there may be no adequate means to express the function in a simple manner without resorting to complex numbers. For example, consider the algebraic function determined by the equation Using the cubic formula, one solution is (the red curve in the accompanying image) There is no way to express this function in terms of real numbers only, even though the resulting function is real-valued on the domain of the graph shown. On a more significant theoretical level, using complex numbers allow one to use the powerful techniques of complex analysis to discuss algebraic functions. In particular, the argument principle can be used to show that any algebraic function is in fact an analytic function, at least in the multiple-valued sense. Formally, let p(x, y) be a complex polynomial in the complex variables x and y. Suppose that x ∈ C is such that the polynomial p(x ,y) of y has n distinct zeros. We shall show 0 0 that the algebraic function is analytic in a neighborhood of x . Choose a system of n non- 0 overlapping discs Δ containing each of these zeros. Then by the argument principle i By continuity, this also holds for all x in a neighborhood of x . In particular, p(x,y) has 0 only one root in Δ, given by the residue theorem: i which is an analytic function. Monodromy Note that the foregoing proof of analyticity derived an expression for a system of n different function elements f(x), provided that x is not a critical point of p(x, y). A i critical point is a point where the number of distinct zeros is smaller than the degree of p, and this occurs only where the highest degree term of p vanishes, and where the discriminant vanishes. Hence there are only finitely many such points c , ..., c . 1 m A close analysis of the properties of the function elements f near the critical points can be i used to show that the monodromy cover is ramified over the critical points (and possibly the point at infinity). Thus the entire function associated to the f has at worst algebraic i poles and ordinary algebraic branchings over the critical points. Note that, away from the critical points, we have since the f are by definition the distinct zeros of p. The monodromy group acts by i permuting the factors, and thus forms the monodromy representation of the Galois group of p. (The monodromy action on the universal covering space is related but different notion in the theory of Riemann surfaces.)