Verblunsky theorem

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21: 27.12 Asymptotic Formulas: Primes

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Prime Number Theorem

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22: 1.10 Functions of a Complex Variable

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Picard’s Theorem

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§1.10(iv) Residue Theorem

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Rouché’s Theorem

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Lagrange Inversion Theorem

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Extended Inversion Theorem

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23: 1.4 Calculus of One Variable

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Mean Value Theorem

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Fundamental Theorem of Calculus

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First Mean Value Theorem

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Second Mean Value Theorem

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§1.4(vi) Taylor’s Theorem for Real Variables

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24: 35.2 Laplace Transform

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Convolution Theorem

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25: 24.10 Arithmetic Properties

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§24.10(i) Von Staudt–Clausen Theorem

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26: 1.6 Vectors and Vector-Valued Functions

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Green’s Theorem

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Stokes’s Theorem

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Gauss’s (or Divergence) Theorem

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Green’s Theorem (for Volume)

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27: 14.18 Sums

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§14.18(i) Expansion Theorem

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§14.18(ii) Addition Theorems

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28: 23.20 Mathematical Applications

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§23.20(ii) Elliptic Curves

… ►The geometric nature of this construction is illustrated in McKean and Moll (1999, §2.14), Koblitz (1993, §§6, 7), and Silverman and Tate (1992, Chapter 1, §§3, 4): each of these references makes a connection with the addition theorem (23.10.1). … ► $K$ always has the form $T\times {\mathbb{Z}}^r$ (Mordell’s Theorem: Silverman and Tate (1992, Chapter 3, §5)); the determination of $r$, the rank of $K$, raises questions of great difficulty, many of which are still open. …

29: Errata

… ►We now include Markov’s Theorem. In regard to orthogonal polynomials on the unit circle, we now discuss monic polynomials, Verblunsky’s Theorem, and Szegő’s theorem. … ►

The largest known prime, which is a Mersenne prime, was updated from $2^{43,112,609}-1$ (2009) to $2^{82,589,933}-1$ (2018).

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33.14.15 $${\int }_0^{\mathrm{\infty }}{\varphi }_{m,\mathrm{\ell }}(r){\varphi }_{n,\mathrm{\ell }}(r)dr={\delta }_{m,n}$$

The definite integral, originally written as ${\int }_0^{\mathrm{\infty }}{\varphi }_{n,\mathrm{\ell }}^2(r)dr=1$, was clarified and rewritten as an orthogonality relation. This follows from (33.14.14) by combining it with Dunkl (2003, Theorem 2.2).

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30: 1.12 Continued Fractions

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Pringsheim’s Theorem

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Van Vleck’s Theorem

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