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1: 8 Incomplete Gamma and Related
Functions
Chapter 8 Incomplete Gamma and Related Functions
…2: 1.2 Elementary Algebra
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βΊThe matrix has a determinant, , explored further in §1.3, denoted, in full index form, as
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βΊIf det()
, has a unique inverse, , such that
…A square matrix is singular if , otherwise it is non-singular.
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βΊIf the matrices and are said to commute.
The difference between and is the commutator denoted as
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3: 3.2 Linear Algebra
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βΊAssume that can be factored as in (3.2.5), but without partial pivoting.
…
βΊis called the characteristic polynomial of and its zeros are the eigenvalues of .
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βΊLet
be an symmetric matrix.
…has the same eigenvalues as .
…
βΊMany methods are available for computing eigenvalues; see Golub and Van Loan (1996, Chapters 7, 8), Trefethen and Bau (1997, Chapter 5), and Wilkinson (1988, Chapters 8, 9).
4: Bibliography H
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βΊ
Certain integrals that contain a probability function.
Bul. Akad. Ε tiince RSS Moldoven. 1975 (2), pp. 86–88, 95 (Russian).
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βΊ
Solving Ordinary Differential Equations. I. Nonstiff Problems.
2nd edition, Springer Series in Computational Mathematics, Vol. 8, Springer-Verlag, Berlin.
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βΊ
Lamé polynomials of large order.
SIAM J. Math. Anal. 8 (5), pp. 800–842.
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βΊ
Calculation of Coulomb wave functions for high energies.
Phys. Rev. 54 (8), pp. 627–628.
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βΊ
On congruences involving Bernoulli numbers and irregular primes. II.
Rep. Fac. Sci. Technol. Meijo Univ. 31, pp. 1–8.
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5: Bibliography
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βΊ
Tables of for Complex Argument.
Pergamon Press, New York.
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βΊ
Supplement to a paper “On the intensity of light in the neighbourhood of a caustic”.
Trans. Camb. Phil. Soc. 8, pp. 595–599.
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βΊ
Frequency response characteristics of the multiport planar elliptic patch.
IEEE Trans. Microwave Theory Tech. 40 (8), pp. 1726–1730.
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βΊ
Algorithm 588. Fast Hankel transforms using related and lagged convolutions.
ACM Trans. Math. Software 8 (4), pp. 369–370.
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βΊ
Chiral-symmetry breaking in QCD. I. The infrared domain.
Phys. Rev. D (3) 37 (8), pp. 2290–2295.
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6: 21.5 Modular Transformations
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βΊLet , , , and
be
matrices with integer elements such that
…Here is an eighth root of unity, that is, .
For general , it is difficult to decide which root needs to be used.
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βΊ( invertible with integer elements.)
…For a matrix we define , as a column vector with the diagonal entries as elements.
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7: Bibliography U
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βΊ
On Kelvin’s ship-wave pattern.
J. Fluid Mech. 8 (3), pp. 418–431.
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βΊ
Integrals with a large parameter: Legendre functions of large degree and fixed order.
Math. Proc. Cambridge Philos. Soc. 95 (2), pp. 367–380.
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8: 29.7 Asymptotic Expansions
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βΊ
29.7.6
βΊ
29.7.7
βΊ
29.7.8
βΊFormulas for additional terms can be computed with the author’s Maple program LA5; see §29.22.
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9: 1.12 Continued Fractions
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βΊ
and are called the th (canonical) numerator and denominator respectively.
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βΊDefine
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βΊA sequence in the extended complex plane, , can be a sequence of convergents of the continued fraction (1.12.3) iff
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βΊConversely, is called an extension of .
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βΊThen the convergents satisfy
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