# Tauberian theorems

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Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)). …
##### 3: 27.15 Chinese Remainder Theorem
###### §27.15 Chinese Remainder Theorem
The Chinese remainder theorem states that a system of congruences $x\equiv a_{1}\pmod{m_{1}},\dots,x\equiv a_{k}\pmod{m_{k}}$, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod $m$), where $m$ is the product of the moduli. This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod $m_{1}$), (mod $m_{2}$), (mod $m_{3}$), and (mod $m_{4}$), respectively. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $\pmod{m}$, which is correct to 20 digits. …
##### 4: 30.10 Series and Integrals
For an addition theorem, see Meixner and Schäfke (1954, p. 300) and King and Van Buren (1973). …
##### 5: 19.35 Other Applications
###### §19.35(i) Mathematical
Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute $\pi$ to high precision (Borwein and Borwein (1987, p. 26)). …
##### 7: 14.28 Sums
For generalizations in terms of Gegenbauer and Jacobi polynomials, see Theorem 2. 1 in Cohl (2013b) and Theorem 1 in Cohl (2013a) respectively. …
##### 8: 13.13 Addition and Multiplication Theorems
###### §13.13(ii) Addition Theorems for $U\left(a,b,z\right)$
13.13.12 $e^{y}\left(\frac{x+y}{x}\right)^{1-b}\sum_{n=0}^{\infty}\frac{(-y)^{n}}{n!x^{n% }}U\left(a-n,b-n,x\right),$ $|y|<|x|$.
##### 9: 10.23 Sums
###### §10.23(i) Multiplication Theorem
Thus, $y\equiv x^{r}\pmod{n}$ and $1\leq y. … By the Euler–Fermat theorem (27.2.8), $x^{\phi\left(n\right)}\equiv 1\pmod{n}$; hence $x^{t\phi\left(n\right)}\equiv 1\pmod{n}$. …