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Uniform (very) sharp bounds for ratios of parabolic cylinder functions
Author(s) -
Segura Javier
Publication year - 2021
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12401
Subject(s) - mathematics , monotonic function , cylinder , elementary function , trigonometric functions , parabolic cylinder function , simple (philosophy) , trigonometry , algebraic number , mathematical analysis , upper and lower bounds , geometry , parabolic partial differential equation , partial differential equation , philosophy , epistemology
Parabolic cylinder functions are classical special functions with applications in many different fields. However, there is little information available regarding simple uniform approximations and bounds for these functions. We obtain very sharp bounds for the ratioΦ n ( x ) = U ( n − 1 , x ) / U ( n , x )and the double ratioΦ n ( x ) / Φ n + 1( x )in terms of elementary functions (algebraic or trigonometric) and prove the monotonicity of these ratios; bounds for U ( n , z ) / U ( n , y ) are also made available. The bounds are very sharp as x → ± ∞ and n → + ∞ , and this simultaneous sharpness in three different directions explains their remarkable global accuracy. Upper and lower elementary bounds are obtained which are able to produce several digits of accuracy for moderately large | x | and/or n .