Analytic continuation of hypergeometric functions by complex single loop Euler transforms

1972 ◽  
Vol 5 (2) ◽  
pp. 256-262 ◽  
Author(s):  
D S F Crothers
Keyword(s):  
Analytic Continuation ◽  
Hypergeometric Functions ◽  
Single Loop

scholarly journals Uniform asymptotic expansions for the Whittaker functions M κ , μ ( z ) and W κ , μ ( z ) with μ large

2021 ◽  
Vol 477 (2252) ◽  
pp. 20210360
Author(s):  
T. M. Dunster
Keyword(s):  
Analytic Continuation ◽  
Error Bounds ◽  
Asymptotic Expansions ◽  
Hypergeometric Functions ◽  
Whittaker Functions ◽  
Airy Functions ◽  
Elementary Functions ◽  
Confluent Hypergeometric Functions ◽  
Uniform Asymptotic Expansions

Uniform asymptotic expansions are derived for Whittaker’s confluent hypergeometric functions M κ , μ ( z ) and W κ , μ ( z ) , as well as the numerically satisfactory companion function W − κ , μ ( z   e − π i ) . The expansions are uniformly valid for μ → ∞ , 0 ≤ κ / μ ≤ 1 − δ < 1 and 0 ≤ arg ⁡ ( z ) ≤ π . By using appropriate connection and analytic continuation formulae, these expansions can be extended to all unbounded non-zero complex z . The approximations come from recent asymptotic expansions involving elementary functions and Airy functions, and explicit error bounds are either provided or available.

scholarly journals THE SCALAR BOX INTEGRAL AND THE MELLIN–BARNES REPRESENTATION

2011 ◽  
Vol 26 (15) ◽  
pp. 2557-2568 ◽  
Author(s):  
P. VALTANCOLI
Keyword(s):  
Analytic Continuation ◽  
Hypergeometric Functions ◽  
Laurent Expansion

We solve exactly the scalar box integral using the Mellin–Barnes representation. First we recognize the hypergeometric functions resumming the series coming from the scalar integrals, then we perform an analytic continuation before applying the Laurent expansion in ϵ = (d-4)/2 of the result.

scholarly journals Some expansions of hypergeometric functions in series of hypergeometric functions

1976 ◽  
Vol 17 (1) ◽  
pp. 17-21 ◽  
Author(s):  
H. M. Srivastava ◽  
Rekha Panda
Keyword(s):  
Analytic Continuation ◽  
Hypergeometric Functions ◽  
Present Note ◽  
Expansion Formula ◽  
In Series ◽  
Image Position

Throughout the present note we abbreviate the set of p parameters a1,…,ap by (ap), with similar interpretations for (bq), etc. Also, by [(ap)]m we mean the product , where [λ]m = Г(λ + m)/ Г(λ), and so on. One of the main results we give here is the expansion formula(1)which is valid, by analytic continuation, when, p,q,r,s,t and u are nonnegative integers such that p+r < q+s+l (or p+r = q+s+l and |zω| <1), p+t < q+u (or p + t = q + u and |z| < 1), and the various parameters including μ are so restricted that each side of equation (1) has a meaning.

Integral representations for solutions of Heun's equation

1969 ◽  
Vol 65 (2) ◽  
pp. 447-459 ◽  
Author(s):  
B. D. Sleeman
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Analytic Continuation ◽  
Hypergeometric Functions ◽  
Integral Representations ◽  
Contour Integrals ◽  
Heun Functions ◽  
Homogeneous Integral Equation ◽  
Heun’S Equation ◽  
Complicated Nature

AbstractIn 1914 Whittaker(12) conjectured that the Heun differential equation is the simplest equation of Fuchsian type whose solution cannot be represented by a contour integral; instead the nearest approach to such a solution is to find a homogeneous integral equation satisfied by a solution of the differential equation. In this paper we reconsider Whittaker's conjecture and show that in fact solutions of Heun's equation can be represented in terms of contour integrals, similar to those of Barnes for the hypergeometric equation. The integrands of these integrals are of a rather complicated nature and cannot be said to involve known or simpler functions although they do provide expressions for the analytic continuation of Heun functions analogous to those for the hypergeometric functions.

A comparative study on the demodulation capability of a single loop optoelectronic oscillator with and without a dynamic band Pass filter

Optik ◽  
2021 ◽  
Vol 226 ◽  
pp. 165924
Author(s):  
Shantanu Mandal ◽  
Kousik Bishayee ◽  
Arindum Mukherjee ◽  
B N Biswas ◽  
Chandan Kumar Sarkar
Keyword(s):  
Comparative Study ◽  
Band Pass Filter ◽  
Optoelectronic Oscillator ◽  
Pass Filter ◽  
Single Loop ◽  
Band Pass

Mobility of Single-Loop N-Bar Linkage With Active/Passive Prismatic Joints

2005 ◽  
Vol 128 (6) ◽  
pp. 1261-1271 ◽  
Author(s):  
W. Z. Guo ◽  
R. Du
Keyword(s):  
Open Loop ◽  
Class Iii ◽  
Active Joint ◽  
Revolute Joint ◽  
Single Loop ◽  
Prismatic Joint ◽  
Passive Joint ◽  
Offset Distance ◽  
Prismatic Joints ◽  
Planar Linkages

Single-loop N-bar linkages that contain one prismatic joint are common in engineering. This type of mechanism often requires complicated control and, hence, understanding its mobility is very important. This paper presents a systematic study on the mobility of this type of mechanism by introducing the concept of virtual link. It is found that this type of mechanism can be divided into three categories: Class I, Class II, and Class III. For each category, the slide reachable range is cut into different regions: Grashof region, non-Grashof region, and change-point region. In each region, the rotation range of the revolute joint or rotatability of the linkage can be determined based on Ting’s criteria. The characteristics charts are given to describe the rotatability condition. Furthermore, if the prismatic joint is an active joint, the revolvability of the input revolute joint is dependent in non-Grashof region but independent in other regions. If the prismatic joint is a passive joint, the revolvability of the input revolute joint is dependent on the offset distance of the prismatic joint. Two examples are given to demonstrate the presented method. The new method is able to cover all the cases of N-bar planar linkages with one or a set of adjoined prismatic joints. It can also be used to study N-bar open-loop planar robotic mechanisms.

scholarly journals Hadamard k-fractional inequalities of Fejér type for GA-s-convex mappings and applications

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hui Lei ◽  
Gou Hu ◽  
Zhi-Jie Cao ◽  
Ting-Song Du
Keyword(s):  
Lower Bounds ◽  
Hypergeometric Functions ◽  
Upper And Lower Bounds ◽  
Weighted Inequalities ◽  
Convex Mappings ◽  
Special Means

Abstract The main aim of this paper is to establish some Fejér-type inequalities involving hypergeometric functions in terms of GA-s-convexity. For this purpose, we construct a Hadamard k-fractional identity related to geometrically symmetric mappings. Moreover, we give the upper and lower bounds for the weighted inequalities via products of two different mappings. Some applications of the presented results to special means are also provided.

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