Computation of the Binet and Gamma Functions by Stieltjes Continued Fractions

One of the most important functions in mathematics and mathematical physics is the gamma function Γ(z). Among several equivalent definitions of Γ(z), we state the following (1.1) Γ ( z ) : = 2 π     z z − ( 1 / 2 ) e − z e J ( z ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072126/15070696-a94f-4da5-a117-8ee069a896b3/content/eq1787.tif"/> which is given in terms of the Binet function J(z) defined by (1.2a) J ( z ) z : = ∫ 0 ∞ υ ( t ) z + t   d t = : G J ( z ) , for z ∈ S π , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072126/15070696-a94f-4da5-a117-8ee069a896b3/content/eq1788.tif"/> where (1.2b) υ ( t ) : = 1 2 π 1 t Log 1 1 − e − 2 π t ,               0 < t < ∞ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072126/15070696-a94f-4da5-a117-8ee069a896b3/content/eq1789.tif"/> and (1.2C) S θ : = [ z ∈ ℂ :   | arg z |   < θ ] ,               0 ≤ θ ≤ π . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072126/15070696-a94f-4da5-a117-8ee069a896b3/content/eq1790.tif"/>