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 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 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.

scholarly journals The a-points of Faber polynomials for a special function

1970 ◽  
Vol 11 (1) ◽  
pp. 1-6
Author(s):  
Hassoon S. Al-Amiri
Keyword(s):  
Analytic Function ◽  
Power Series ◽  
Analytic Continuation ◽  
Special Function ◽  
Laurent Expansion ◽  
Faber Polynomials ◽  
Analytic Element ◽  
Formal Expansion ◽  
Image Position

Let f(ζ) be a power series of the formwhere lim sup |an|1/n < ∞. The Faber polynomials {fn(ζ)} (n = 0, 1, 2, …) are the polynomial parts of the formal expansion of (f(ζ))n about ζ = ∞. Series (1) defines an analytic element of an analytic function which we designate as w = f(ζ). Since at ζ = ∞ the analytic element is univalent in some neighborhood of infinity; thus the inverse of this element is uniquely determined in some neighborhood of w= ∞, and has a Laurent expansion of the formwhere lim sup |bn|1/n = p < ∞. Let ζ = g(w) be this single-valued function defined by (2) in |w| > p. No analytic continuation of g(w) will be considered.

scholarly journals On certain sums over ordinates of zeta-zeros II

2019 ◽  
Author(s):  
Andriy Bondarenko ◽  
Aleksandar Ivić ◽  
Eero Saksman ◽  
Kristian Seip
Keyword(s):  
Analytic Continuation ◽  
Critical Line ◽  
Laurent Expansion ◽  
Second Moment ◽  
International Audience ◽  
Complex Zeros ◽  
Gamma S

International audience Let γ denote the imaginary parts of complex zeros ρ = β + iγ of ζ(s). The problem of analytic continuation of the function $G(s) :=\sum_{\gamma >0} {\gamma}^{-s}$ to the left of the line $\Re{s} = −1 $ is investigated, and its Laurent expansion at the pole s = 1 is obtained. Estimates for the second moment on the critical line $\int_{1}^{T} {| G (\frac{1}{2} + it) |}^2 dt $ are revisited. This paper is a continuation of work begun by the second author in [Iv01].

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.

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