Finite endpoint power substitution followed by Gauss–Jacobi quadrature

Assume Countable Choice. Input $a<b$, an integer $n\ge 1$, exponents $\alpha ,\beta >-1$, nonnegative integers $m_a,m_b$, a function $h\in C^{2n}([a,b])$ in the sense of Higher derivatives and the classes $C^k$ and $C^{\mathrm{\infty }}$, and positive integers $q_a,q_b$ satisfying $q_a(\alpha +1)-1>2n$ and $q_b(\beta +1)-1>2n$. On $(a,b)$ define $f(x)=(x-a{)}^{\alpha }(b-x{)}^{\beta }{(\log \frac{x-a}{b-a})}^{m_a}{(\log \frac{b-x}{b-a})}^{m_b}h(x)\mathrm{.}$ The logarithms are natural logarithms and their arguments are positive. Put $c=(a+b)\mathrm{/}2$, $x_L(t)=a+(c-a)t^{q_a}$, and $x_R(t)=b-(b-c)t^{q_b}$. For $0<t\le 1$ set $F_L(t)=f(x_L(t))q_a(c-a)t^{q_a-1}$ and $F_R(t)=f(x_R(t))q_b(b-c)t^{q_b-1}$; set $F_L(0)=F_R(0)=0$.

Let $(z_j,w_j{)}_{j=1}^n$ be the nodes and weights of the $n$-node Gauss–Jacobi rule for $\alpha =\beta =0$ on $[-1,1]$. Set $s_j=(1+z_j)\mathrm{/}2$ and $v_j=w_j\mathrm{/}2$. The output is the finite number $\widehat{I}={\sum }_{j=1}^n{v}_j{F}_L(s_j)+{\sum }_{j=1}^n{v}_j{F}_R(s_j)\mathrm{.}$ The inputs $q_a,q_b$ are supplied finite cutoff orders; this definition makes no claim of automatic parameter selection. The conditional integral and error guarantee are proved in Error bound after endpoint power regularization ↗.