Neumann%20addition%20theorem

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1: 10.44 Sums

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§10.44(i) Multiplication Theorem

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§10.44(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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§10.44(iii) Neumann-Type Expansions

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2: 10.23 Sums

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Neumann’s Addition Theorem

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Neumann’s Expansion

… ►and $O_k\left(t\right)$ is Neumann’s polynomial, defined by the generating function: … ► $O_n\left(t\right)$ is a polynomial of degree $n+1$ in $1/t:O_0\left(t\right)=1/t$ and …

3: 20 Theta Functions

Chapter 20 Theta Functions

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4: 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)). …

5: Tom M. Apostol

… ►Apostol was born on August 20, 1923. … ►In 1998, the Mathematical Association of America (MAA) awarded him the annual Trevor Evans Award, presented to authors of an exceptional article that is accessible to undergraduates, for his piece entitled “What Is the Most Surprising Result in Mathematics?” (Answer: the prime number theorem). …In addition, he was the co-author of New Horizons in Geometry, published by the MAA, which received the CHOICE “Outstanding Academic Title” award in 2013. …He additionally served as a visiting lecturer for the MAA, and as a member of the MAA Board of Governors. …

6: 27.15 Chinese Remainder Theorem

§27.15 Chinese Remainder Theorem

… ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …Their product $m$ has 20 digits, twice the number of digits in the data. 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. …

7: 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. … ►

20 Theta Functions

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8: Errata

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Equation (5.2.9).

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Equation (16.16.5_5).

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10.23.11 $$a_k=\frac{1}{2\pi i}{\int }_{|t|=c^{\prime }}f(t)O_k\left(t\right)dt,$$

Originally the contour of integration written incorrectly as $|z|=c^{\prime }$, has been corrected to be $|t|=c^{\prime }$.

Reported by Mark Dunster on 2021-03-22

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Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.

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9: 8 Incomplete Gamma and Related
Functions

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10: 28 Mathieu Functions and Hill’s Equation

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