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Gauss–Jacobi regularization for x to the minus one half times log x

Statement

The improper integral is $\int_{0}^{1} x^{- 1 / 2} \log x d x = - 4.$ For example, Finite endpoint power substitution followed by Gauss–Jacobi quadrature applies with $n = 1$ , $α = - 1 / 2$ , $β = 0$ , $m_{a} = 1$ , $m_{b} = 0$ , $h (x) = 1$ , $q_{a} = 7$ , and $q_{b} = 4$ . The transformed functions are $C^{2}$ and Error bound after endpoint power regularization gives an explicit error bound $C_{1} (∥ F_{L}^{''} ∥_{\infty} + ∥ F_{R}^{''} ∥_{\infty})$ for the returned two-node sum.

Facts & Assumptions

Given: The integrand $f (x) = x^{- 1 / 2} \log x$ on $(0, 1]$ and the specified endpoint-regularization inputs.

[F1]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: $\log$ is differentiable on $(0, \infty)$ with derivative $1 / x$ .

[F2]

A sufficiently high power regularizes finite logarithmic endpoint factors: $t^{δ} P (\log t)$ extends to $C^{r} ([0, 1])$ when $δ > r$ and $P$ is a polynomial.

[F3]

Finite endpoint power substitution followed by Gauss–Jacobi quadrature: the left and right substitution orders $q_{a}, q_{b}$ and the shifted Gauss–Jacobi sums define the stated finite output.

[F4]

Error bound after endpoint power regularization: when the substituted functions are $C^{2 n}$ , the improper integral exists and the error is bounded by the displayed derivative constant.

[F5]

The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ : on $[ε, 1]$ , the integral of a derivative equals the primitive's endpoint difference.

[F6]

Continuity and derivatives of positive-base real powers, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , and The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ : real powers, products, and compositions have the derivatives used for the primitive below.

Proof

1.1F1F2F5F6algebra

A primitive on $(0, 1]$ is $P (x) = 2 \sqrt{x} \log x - 4 \sqrt{x}$ by [F1] and [F6]. The endpoint term $\sqrt{x} \log x$ tends to zero by the power-regularity lemma [F2] with $r = 0$ , $δ = 1 / 2$ , and polynomial $z$ . Hence $P (x) \to 0$ as $x ↓ 0$ . Applying [F5] on $[ε, 1]$ and taking $ε ↓ 0$ gives the improper integral $P (1) - 0 = - 4$ .

1.2F2F3givenalgebra

For the left substitution, $δ_{L} = q_{a} (α + 1) - 1 = 7 / 2 - 1 = 5 / 2 > 2$ ; its transformed function is a constant times $t^{5 / 2}$ times a polynomial of degree one in $\log t$ and a smooth factor bounded with two continuous derivatives. For the right substitution, $δ_{R} = q_{b} (β + 1) - 1 = 3 > 2$ and the remaining factors are smooth on $[0, 1]$ . Therefore both transformed functions are $C^{2}$ by [F2], and [F3] applies.

2.1F3F4step 1.1step 1.2∎

Apply [F4] with $n = 1$ to the functions in step 1.2. Its error constant is $C_{1}$ and the output is the stated sum, so the claimed error bound follows. The value $- 4$ from step 1.1 is the integral being approximated.

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Sources

J. Lebl, Basic Analysis I, §5.4.1–5.4.2, The logarithm and the exponential (standard reference, not scraped)
Christopher J. Bishop, The Riemann Mapping Theorem, Chapter 3 §3, Gauss-Jacobi quadrature lecture notes (standard reference, not scraped)