# Raabe theorem

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##### 1: 28.27 Addition Theorems

###### §28.27 Addition Theorems

►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)). …##### 2: 24.4 Basic Properties

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###### Raabe’s Theorem

…##### 3: 27.15 Chinese Remainder Theorem

###### §27.15 Chinese Remainder Theorem

►The Chinese remainder theorem states that a system of congruences $x\equiv {a}_{1}\phantom{\rule{veryverythickmathspace}{0ex}}(mod{m}_{1}),\mathrm{\dots},x\equiv {a}_{k}\phantom{\rule{veryverythickmathspace}{0ex}}(mod{m}_{k})$, 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 $(modm)$, which is correct to 20 digits. …##### 4: 24.1 Special Notation

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►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851.
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##### 5: 30.10 Series and Integrals

##### 6: 19.35 Other Applications

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###### §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: 10.44 Sums

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###### §10.44(i) Multiplication Theorem

… ►###### §10.44(ii) Addition Theorems

►###### Neumann’s Addition Theorem

… ►###### Graf’s and Gegenbauer’s Addition Theorems

…##### 8: 13.13 Addition and Multiplication Theorems

###### §13.13 Addition and Multiplication Theorems

►###### §13.13(i) Addition Theorems for $M(a,b,z)$

… ►###### §13.13(ii) Addition Theorems for $U(a,b,z)$

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13.13.12
$${\mathrm{e}}^{y}{\left(\frac{x+y}{x}\right)}^{1-b}\sum _{n=0}^{\mathrm{\infty}}\frac{{(-y)}^{n}}{n!{x}^{n}}U(a-n,b-n,x),$$
$$.

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###### §13.13(iii) Multiplication Theorems for $M(a,b,z)$ and $U(a,b,z)$

…##### 9: 10.23 Sums

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