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21: 12.16 Mathematical Applications

… ►PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turning points; see §§2.4(vi) and 2.8(vi). …

22: Peter L. Walker

… ►Walker’s published work has been mainly in real and complex analysis, with excursions into analytic number theory and geometry, the latter in collaboration with Professor Mowaffaq Hajja of the University of Jordan. ►Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. …

23: Wadim Zudilin

… ►His research interests are primarily focused on applications of special functions in different parts of number theory. …He is a member of several editorial boards including the series Monographs in Number Theory published by World Scientific. …

24: 14.31 Other Applications

… ►

§14.31(ii) Conical Functions

… ►These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). …

25: 27.9 Quadratic Characters

§27.9 Quadratic Characters

►For an odd prime

p

, the Legendre symbol

(n|p)

is defined as follows. … ►If

p, q

are distinct odd primes, then the quadratic reciprocity law states that … ►If an odd integer

P

has prime factorization

P=\prod_{r=1}^{\nu\left(n\right)}p^{a_{r}}_{r}

, then the Jacobi symbol

(n|P)

is defined by

(n|P)=\prod_{r=1}^{\nu\left(n\right)}{(n|p_{r})}^{a_{r}}

, with

(n|1)=1

. …

26: 27.13 Functions

… ►

§27.13(i) Introduction

►Whereas multiplicative number theory is concerned with functions arising from prime factorization, additive number theory treats functions related to addition of integers. …The subsections that follow describe problems from additive number theory. … ►

§27.13(ii) Goldbach Conjecture

… ►

§27.13(iii) Waring’s Problem

…

27: 27.8 Dirichlet Characters

§27.8 Dirichlet Characters

… ►An example is the principal character (mod

k

): … ►If

\left(n,k\right)=1

, then the characters satisfy the orthogonality relation … ►A divisor

d

k

is called an induced modulus for

\chi

if … ►If

k

is odd, then the real characters (mod

k

) are the principal character and the quadratic characters described in the next section.

28: 27.14 Unrestricted Partitions

… ►Euler’s pentagonal number theorem states that … ►and

s(h,k)

is a Dedekind sum given by … ►For example, the Ramanujan identity … ►

§27.14(vi) Ramanujan’s Tau Function

… ►

29: Sidebar 9.SB1: Supernumerary Rainbows

… ►Airy invented his function in 1838 precisely to describe this phenomenon more accurately than Young had done in 1800 when pointing out that supernumerary rainbows require the wave theory of light and are impossible to explain with Newton’s picture of light as a stream of independent corpuscles. …

30: Antony Ross Barnett

… ►Barnett’s research interests included number theory and special functions. …