Floquet theorem

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11: 10.23 Sums

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§10.23(i) Multiplication Theorem

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§10.23(ii) Addition Theorems

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Neumann’s Addition Theorem

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Graf’s and Gegenbauer’s Addition Theorems

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12: 27.16 Cryptography

… ►Thus, $y\equiv x^r(modn)$ and . … ►By the Euler–Fermat theorem (27.2.8), $x^{\varphi \left(n\right)}\equiv 1(modn)$; hence $x^{t\varphi \left(n\right)}\equiv 1(modn)$. …

13: 14.28 Sums

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§14.28(i) Addition Theorem

… ►For generalizations in terms of Gegenbauer and Jacobi polynomials, see Theorem 2. 1 in Cohl (2013b) and Theorem 1 in Cohl (2013a) respectively. …

14: 13.26 Addition and Multiplication Theorems

§13.26 Addition and Multiplication Theorems

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§13.26(i) Addition Theorems for $M_{\kappa ,\mu }\left(z\right)$

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§13.26(ii) Addition Theorems for $W_{\kappa ,\mu }\left(z\right)$

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§13.26(iii) Multiplication Theorems for $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$

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15: 28.12 Definitions and Basic Properties

… ►In consequence, for the Floquet solutions $w(z)$ the factor ${\mathrm{e}}^{\pi \mathrm{i}\nu }$ in (28.2.14) is no longer $\pm 1$. … ►The Floquet solution with respect to $\nu$ is denoted by ${\mathrm{me}}_{\nu }(z,q)$. …The other eigenfunction is ${\mathrm{me}}_{\nu }(-z,q)$, a Floquet solution with respect to $-\nu$ with $a={\lambda }_{\nu }\left(q\right)$. …

16: 22.18 Mathematical Applications

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§22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem

… ►With the identification $x=\mathrm{sn}(z,k)$, $y=d(\mathrm{sn}(z,k))/dz$, the addition law (22.18.8) is transformed into the addition theorem (22.8.1); see Akhiezer (1990, pp. 42, 45, 73–74) and McKean and Moll (1999, §§2.14, 2.16). …

17: Hans Volkmer

… ►His book Multiparameter Eigenvalue Problems and Expansion Theorems was published by Springer as Lecture Notes in Mathematics No. …

18: Peter L. Walker

… ►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. …

19: 12.13 Sums

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§12.13(i) Addition Theorems

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20: 27.2 Functions

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§27.2(i) Definitions

… ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) ►This result, first proved in Hadamard (1896) and de la Vallée Poussin (1896a, b), is known as the prime number theorem. …This is the number of positive integers $\le n$ that are relatively prime to $n$; $\varphi \left(n\right)$ is Euler’s totient. ►If $\left(a,n\right)=1$, then the Euler–Fermat theorem states that …