Fast computation of complete elliptic integrals and Jacobian elliptic functions

2009 ◽  
Vol 105 (4) ◽  
pp. 305-328 ◽  
Author(s):  
Toshio Fukushima
Keyword(s):  
Elliptic Functions ◽  
Elliptic Integrals ◽  
Fast Computation ◽  
Jacobian Elliptic Functions ◽  
Complete Elliptic Integrals

scholarly journals HYPERBOLIC WAVES AND THEIR SHOALING

1968 ◽  
Vol 1 (11) ◽  
pp. 9 ◽  
Author(s):  
Yuichi Iwagaki
Keyword(s):  
Wave Height ◽  
Elliptic Functions ◽  
Wave Theory ◽  
Wave Crest ◽  
Cnoidal Wave ◽  
Practical Application ◽  
Jacobian Elliptic Functions ◽  
Complete Elliptic Integrals ◽  
Hyperbolic Waves ◽  
The Waves

Is is very difficult for engineers to deal with the cnoidal wave theory for practical application, since this theory contains the Jacobian elliptic functions, their modulus k, and the complete elliptic integrals of the first and second kinds, K and E respectively. This paper firstly proposes formulae for various wave characteristics of new waves named "hyperbolic waves", which are derived from the cnoidal wave theory under the condition that k = 1 and E = 1 but K is not infinite and are a function of T/g/h and H/h, so that cnoidal waves can be approximately expressed as hyperbolic waves by primary functions only, in which T is the wave period, h the water depth and H the wave height. Secondly, as an application of the hyperbolic wave theory, the present paper deals with wave shoaling, that is, changes in the wave height, the wave crest height above still water level, and the wave velocity, when the waves proceed into shallow water from deep water.

scholarly journals Master integrals for bipartite cuts of three-loop photon self energy

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
R. N. Lee ◽  
A. I. Onishchenko
Keyword(s):  
Spectral Density ◽  
High Energy ◽  
Elliptic Integrals ◽  
Iterated Integrals ◽  
Self Energy ◽  
Complete Elliptic Integrals ◽  
The One

Abstract We calculate the master integrals for bipartite cuts of the three-loop propagator QED diagrams. These master integrals determine the spectral density of the photon self energy. Our results are expressed in terms of the iterated integrals, which, apart from the 4m cut (the cut of 4 massive lines), reduce to Goncharov’s polylogarithms. The master integrals for 4m cut have been calculated in our previous paper in terms of the one-fold integrals of harmonic polylogarithms and complete elliptic integrals. We provide the threshold and high-energy asymptotics of the master integrals found, including those for 4m cut.

Jacobian elliptic functions and a family of bivariate peak polynomials

2021 ◽  
Vol 97 ◽  
pp. 103371
Author(s):  
Shi-Mei Ma ◽  
Jun Ma ◽  
Yeong-Nan Yeh ◽  
Roberta R. Zhou
Keyword(s):  
Elliptic Functions ◽  
Jacobian Elliptic Functions
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