By Woerle K.
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Additional resources for 200 Aufgaben aus der Trigonometrie mit Loesungen
Hence 1 ξ(s) ≡ π −s/2 Γ ( s)ζ(s) = 2 1 = 0 s/2 x 1 0 xs/2 Ψ (x)dx + 1 1 1 1 √ Ψ( ) + √ − x x 2 x 2 ∞ 1 xs/2 Ψ (x)dx = ∞ dx + 1 xs/2 Ψ (x)dx 1 ∞ 1 1 1 − + = xs/2−3/2 Ψ ( )dx + xs/2 Ψ (x)dx = s−1 s x 0 1 ∞ 1 = + x−s/2−1/2 + xs/2−1 Ψ (x)dx . s(s − 1) 1 Quantum and Arithmetical Chaos 43 The last integral is convergent for all values of s and gives the analytical continuation of the Riemann zeta function to the whole complex s-plane, the only singularity being the pole at s = 1 with unit residue s→1 ζ(s) −→ 1 .
Comparing these expressions one concludes that the diagonal approximation reproduces the correct small-t behavior of form-factors of classical ensembles (cf. (44)). Unfortunately, K (diag) (t) grows with increasing t but the exact form-factor for systems without spectral degeneracy should tend to d¯ for large t. This is a consequence of the following arguments. According to (45) δ(E − En )δ(E + − Em ) R2 ( ) = m,n δ(E − En )δ( − Em + En ) = . m,n If there is no levels with exactly the same energy the second δ-function in the right hand side of this equation tends to δ( ) when → 0 and the ﬁrst one ¯ Therefore gives d.
G. 1, Sect. 3) (1 − u2 ) d2 G dG + l(l + 1)G = 0 − 2u du2 du where E= 1 + k 2 = −l(l + 1) 4 and 1 l = − − ik . 2 As for the plane case the required solution of the above equation should grow as eikd when d → ∞ and should behave like ln d/2π when d → 0. 1, Sect. 3 it follows that (0) GE (x, x ) = − 1 Q 1 (cosh d(x, x )) . 2π − 2 −ik Here Q− 12 −ik (cosh d) is the Legendre function of the second kind with the integral representation , Vol. 4) 1 Q− 12 −ik (cosh d) = √ 2 ∞ d √ eikr dr cosh r − cosh d and the following asymptotics d→0 Q− 12 −ik (cosh d) −→ − log d and Quantum and Arithmetical Chaos d→∞ Q− 12 −ik (cosh d) −→ 23 π ei(kd−π/4) .
200 Aufgaben aus der Trigonometrie mit Loesungen by Woerle K.