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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 gasflow 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; 1m; (1t)/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+1c; 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 (ca, ca; c; t), 0 < T₁ ≤ t < 1, as –a → ∞, this result then leads to a sharper estimate on the modulus of nth order derivatives of holomorphic functions as n becomes large.
Item Metadata
Title  Asymptotic expansions of the hypergeometric function for large values of the parameters 
Creator  Prinsenberg, Gerard Simon 
Publisher  University of British Columbia 
Date Issued  1966 
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 gasflow 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; 1m; (1t)/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+1c; 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 (ca, ca; c; t), 0 < T₁ ≤ t < 1, as –a → ∞, this result then leads to a sharper estimate on the modulus of nth order derivatives of holomorphic functions as n becomes large.

Subject  Asymptotic expansions; Hypergeometric functions 
Genre  Thesis/Dissertation 
Type  Text 
Language  eng 
Date Available  20110912 
Provider  Vancouver : University of British Columbia Library 
Rights  For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. 
DOI  10.14288/1.0080584 
URI  
Degree  Master of Arts  MA 
Program  Mathematics 
Affiliation  Science, Faculty of; Mathematics, Department of 
Degree Grantor  University of British Columbia 
Campus  UBCV 
Scholarly Level  Graduate 
Aggregated Source Repository  DSpace 
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For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.