Farrell and Ross present a selection of problems, each with a solution worked out in detail, dealing with the properties and applications of the Gamma function and the Beta function
, the Legendre polynomials, and the Bessel functions.
1] (1, b; c; z) = [phi](b; c; z) is known as incomplete beta function
where [beta] (l,m) is the beta function
with l, m > 0 and is defined as
Integration Techniques, Gamma Function, Beta Function, Cauchy
properties of the beta function, rewrite the integral as 1.
Tables of the Incomplete Beta Function
, Cambridge University Press, Cambridge, UK, first edition (1934).
GSK-3 beta function
is important in a variety of biological processes, and mice with the tagged protein exhibited many problems in utero, including cleft palates and sternum defects.
The area of a spherical cap can also be described using the incomplete Beta function
where B is the beta function
and we have used the homogeneity of R to divide the variables by their arithmetic average,
where [alpha] > 0, [beta], > 0, and B([alpha], [beta]) is the beta function
Changing the variable, t = 2x - 1, we get immediately the integral defining the beta function
, and as a consequence
Nine appendices (about 40 pages) give background on mathematical relations, gamma and beta functions
, Fourier series, basic theorems of the theory of second-order and linear second-order differential equations, perturbation methods, and discrete models of two TNL oscillations.