relation to infinite partial fractions
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1: 1.10 Functions of a Complex Variable
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►The convergence of the infinite product is uniform if the sequence of partial products converges uniformly.
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§1.10(x) Infinite Partial Fractions
… ►Mittag-Leffler’s Expansion
… ►The recurrence relation for in §18.9(i) follows from , and the contour integral representation for in §18.10(iii) is just (1.10.27).2: 1.9 Calculus of a Complex Variable
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►If , , then the integral is defined analogously to the infinite integrals in §1.4(v).
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§1.9(v) Infinite Sequences and Series
… ►This sequence converges pointwise to a function if … ► … ►§1.9(vii) Inversion of Limits
…3: 3.10 Continued Fractions
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§3.10(ii) Relations to Power Series
… ►can be converted into a continued fraction of type (3.10.1), and with the property that the th convergent to is equal to the th partial sum of the series in (3.10.3), that is, … ►Stieltjes Fractions
… ►The and of (3.10.2) can be computed by means of three-term recurrence relations (1.12.5). … ►This forward algorithm achieves efficiency and stability in the computation of the convergents , and is related to the forward series recurrence algorithm. …4: 16.7 Relations to Other Functions
§16.7 Relations to Other Functions
…5: 6.11 Relations to Other Functions
§6.11 Relations to Other Functions
… ►Incomplete Gamma Function
… ►Confluent Hypergeometric Function
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6.11.2
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6.11.3
6: 4.22 Infinite Products and Partial Fractions
7: 4.36 Infinite Products and Partial Fractions
8: Bibliography R
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Normal limit theorems for symmetric random matrices.
Probab. Theory Related Fields 112 (3), pp. 411–423.
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Partial fractions expansions and identities for products of Bessel functions.
J. Math. Phys. 46 (4), pp. 043509–1–043509–18.
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The similarity solution for the Korteweg-de Vries equation and the related Painlevé transcendent.
Proc. Roy. Soc. London Ser. A 361, pp. 265–275.
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Sources in the development of mathematics.
Cambridge University Press, Cambridge.
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On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media.
In Differential Operators and Related Topics, Vol. I (Odessa,
1997),
Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.
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9: 15.19 Methods of Computation
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►However, by appropriate choice of the constant in (15.15.1) we can obtain an infinite series that converges on a disk containing .
Moreover, it is also possible to accelerate convergence by appropriate choice of .
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