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Asymptotic expansions of the hypergeometric function for large values of the parameters Prinsenberg, Gerard Simon
Abstract
In chapter I known asymptotic forms and expansions of the hypergeometric function obtained by Erdélyi [5], Hapaev [10,11], Knottnerus [15L Sommerfeld [25] and Watson [28] are discussed. Also the asymptotic expansions of the hypergeometric function occurring in gas-flow theory will be discussed. These expansions were obtained by Cherry [1,2], Lighthill [17] and Seifert [2J]. Moreover, using a paper by Thorne [28] asymptotic expansions of ₂F₁(p+1, -p; 1-m; (1-t)/2), -1 < t < 1, and ₂P₁( (p+m+2)/2, (p+m+1)/2; p+ 3/2-, t⁻² ), t > 1, are obtained as p-»» and m = -(p+ 1/2)a, where a is fixed and 0 < a < 1. The : expansions are in terms of Airy functions of the first kind. The hypergeometric equation is normalized in chapter II. It readily yields the two turning points t₁, i = 1,2. If we consider,the case the a=b is a large real parameter of the hypergeometric function ₂F₁(a,b; c; t), then the turning points coalesce with the regular singularities t = 0 and t = ∞ of the hypergeometric equation as | a | →∞. In chapter III new asymptotic forms are found for this particular case; that is, for ₂F₁ (a, a; c;t) , 0 < T₁ ≤ t < 1, and ₂F₁ (a,a+1-c; 1; t⁻¹), 1 < t ≤ T₂ < ∞ , as –a → ∞ . The asymptotic form is in terms of modified Bessel functions of order 1/2. Asymptotic expansions can be obtained in a similar manner. Furthermore, a new asymptotic form is derived for ₂F₁ (c-a, c-a; c; t), 0 < T₁ ≤ t < 1, as –a → ∞, this result then leads to a sharper estimate on the modulus of n-th order derivatives of holomorphic functions as n becomes large.
Item Metadata
Title |
Asymptotic expansions of the hypergeometric function for large values of the parameters
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
1966
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Description |
In chapter I known asymptotic forms and expansions of the hypergeometric function obtained by Erdélyi [5], Hapaev [10,11], Knottnerus [15L Sommerfeld [25] and Watson [28] are discussed. Also the asymptotic expansions of the hypergeometric function occurring in gas-flow theory will be discussed. These expansions were obtained by Cherry [1,2], Lighthill [17] and Seifert [2J]. Moreover, using a paper by Thorne [28] asymptotic expansions of ₂F₁(p+1, -p; 1-m; (1-t)/2), -1 < t < 1, and ₂P₁( (p+m+2)/2, (p+m+1)/2; p+ 3/2-, t⁻² ), t > 1, are obtained as p-»» and m = -(p+ 1/2)a, where a is fixed and 0 < a < 1. The : expansions are in terms of Airy functions of the first kind.
The hypergeometric equation is normalized in chapter II. It readily yields the two turning points t₁, i = 1,2. If we consider,the case the a=b is a large real parameter of the hypergeometric function ₂F₁(a,b; c; t), then the turning points coalesce with the regular singularities t = 0 and t = ∞ of the hypergeometric equation as | a | →∞.
In chapter III new asymptotic forms are found for this particular case; that is, for
₂F₁ (a, a; c;t) , 0 < T₁ ≤ t < 1, and ₂F₁ (a,a+1-c; 1; t⁻¹), 1 < t ≤ T₂ < ∞ , as –a → ∞ .
The asymptotic form is in terms of modified Bessel functions of order 1/2. Asymptotic expansions can be obtained in a similar manner.
Furthermore, a new asymptotic form is derived for ₂F₁ (c-a, c-a; c; t), 0 < T₁ ≤ t < 1, as –a → ∞, this result then leads to a sharper estimate on the modulus of n-th order derivatives of holomorphic functions as n becomes large.
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Language |
eng
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Date Available |
2011-09-12
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Provider |
Vancouver : University of British Columbia Library
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Rights |
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.
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DOI |
10.14288/1.0080584
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Affiliation | |
Degree Grantor |
University of British Columbia
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Campus | |
Scholarly Level |
Graduate
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Aggregated Source Repository |
DSpace
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For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.