Manuscript Hawk05-B@1 · Hawk05-B.tex · section · f2e5d4f5

The logarithmic integral

Evaluation of the exact Page average › The logarithmic integral · lines 1764–1849

Hawk05-B@1#eq:lagrec, Hawk05-B@1#eq:lagder, Hawk05-B@1#eq:logint

The finite-polynomial or generating-function identities give

x L_{j}^{α} \partial_{α} L_{j}^{α} = (2 j + α + 1) L_{j}^{α} - (j + 1) L_{j + 1}^{α} - (j + α) L_{j - 1}^{α}, = ℓ = 0 \sum^{j - 1} \frac{L _{ℓ} ^{α}}{j - ℓ} .

The first follows by comparing coefficients. For the second, use

j \geq 0 \sum L_{j}^{α} (x) t^{j} = (1 - t)^{- α - 1} exp (- \frac{x t}{1 - t})

and differentiate in $α$ . Terms with index $- 1$ are zero.

Orthogonality and (eq:lagrec) give

A_{j} := \int_{0}^{\infty} x^{α + 1} e^{- x} [L_{j}^{α} (x)]^{2} d x = (2 j + α + 1) h_{j} .

The cross term in the derivative of $A_{j}$ is

\int_{0}^{\infty} x^{α + 1} e^{- x} L_{j}^{α} \partial_{α} L_{j}^{α} d x = - j h_{j} .

Only the $j - 1$ term of (eq:lagder) contributes. Differentiation under the integral is justified for $α > - 1$ , locally uniformly in $α$ , by the exponential weight and integrability of $x^{α} ln x$ . Thus

\frac{1}{h _{j}} \int_{0}^{\infty} x^{α + 1} e^{- x} ln x [L_{j}^{α} (x)]^{2} d x = (2 j + α + 1) ψ_{0} (j + α + 1) + 1 + 2 j .

To sum (eq:logint), put $r = j + α + 1$ and use

ψ_{0} (r) = ψ_{0} (n + 1) - ℓ = r \sum^{n} \frac{1}{ℓ} .

The coefficient sum is

j = 0 \sum^{m - 1} (2 j + α + 1) = mn .

The double sum simplifies because

r = α + 1 \sum^{ℓ} (2 r - α - 1) = ℓ (ℓ - α) .

Consequently

j = 0 \sum^{m - 1} (2 j + α + 1) ψ_{0} (j + α + 1) = mn ψ_{0} (n + 1) - \frac{m ( m + 1 )}{2} .

Since $\sum_{j = 0}^{m - 1} (1 + 2 j) = m^{2}$ , we obtain

E Tr (W ln W) = mn ψ_{0} (n + 1) + \frac{m ( m - 1 )}{2} .

Substitute this expression into (eq:pagegamma):

E S_{nat} (ρ) = ψ_{0} (mn + 1) - ψ_{0} (n + 1) - \frac{m - 1}{2 n} .

Finally $ψ_{0} (ℓ + 1) = H_{ℓ} - γ_{Euler}$ . Division by $ln 2$ proves (eq:page), including $m = 1$ .

Displayed formulas in this passage

Hawk05-B@1#eq:lagrec · line 1767
$x L_{j}^{α} \partial_{α} L_{j}^{α} = (2 j + α + 1) L_{j}^{α} - (j + 1) L_{j + 1}^{α} - (j + α) L_{j - 1}^{α}, = ℓ = 0 \sum^{j - 1} \frac{L _{ℓ} ^{α}}{j - ℓ} .$
Hawk05-B@1#eq:lagder · line 1767
$x L_{j}^{α} \partial_{α} L_{j}^{α} = (2 j + α + 1) L_{j}^{α} - (j + 1) L_{j + 1}^{α} - (j + α) L_{j - 1}^{α}, = ℓ = 0 \sum^{j - 1} \frac{L _{ℓ} ^{α}}{j - ℓ} .$
Hawk05-B@1#eq:logint · line 1802
$\frac{1}{h _{j}} \int_{0}^{\infty} x^{α + 1} e^{- x} ln x [L_{j}^{α} (x)]^{2} d x = (2 j + α + 1) ψ_{0} (j + α + 1) + 1 + 2 j .$
unlabeled · line 1777
$j \geq 0 \sum L_{j}^{α} (x) t^{j} = (1 - t)^{- α - 1} exp (- \frac{x t}{1 - t})$
unlabeled · line 1786
$A_{j} := \int_{0}^{\infty} x^{α + 1} e^{- x} [L_{j}^{α} (x)]^{2} d x = (2 j + α + 1) h_{j} .$
unlabeled · line 1792
$\int_{0}^{\infty} x^{α + 1} e^{- x} L_{j}^{α} \partial_{α} L_{j}^{α} d x = - j h_{j} .$
unlabeled · line 1812
$ψ_{0} (r) = ψ_{0} (n + 1) - ℓ = r \sum^{n} \frac{1}{ℓ} .$
unlabeled · line 1818
$j = 0 \sum^{m - 1} (2 j + α + 1) = mn .$
unlabeled · line 1822
$r = α + 1 \sum^{ℓ} (2 r - α - 1) = ℓ (ℓ - α) .$
unlabeled · line 1827
$j = 0 \sum^{m - 1} (2 j + α + 1) ψ_{0} (j + α + 1) = mn ψ_{0} (n + 1) - \frac{m ( m + 1 )}{2} .$
unlabeled · line 1834
$E Tr (W ln W) = mn ψ_{0} (n + 1) + \frac{m ( m - 1 )}{2} .$
unlabeled · line 1840
$E S_{nat} (ρ) = ψ_{0} (mn + 1) - ψ_{0} (n + 1) - \frac{m - 1}{2 n} .$