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34.7 Other approaches 549The jump coefficients A and B – the scattering data – are algebraicfunctions of β(x) and x. This is a matrix Riemann–Hilbert problem andcan be reformulated as an integral equation. The approach via the axisdata can also be used to derive the Kerr solution (Neugebauer 2000).The solution for the rigidly rotating disc of dust depends on two parameters,the radius ρ 0 and the angular velocity Ω. It exists for µ =2Ω 2 ρ 2 0 e−2V 0≤ µ 0 =4.62966 ... . The surface potential e −2V 0isanimplicitfunction of µ. Forµ = µ 0 the solution reduces to the extreme Kerrsolution.The solution of the Riemann–Hilbert problem leads to a class ofErnst potentials involving elliptic and hyperelliptic functions (Meinel andNeugebauer 1996). Let, for given n, K i ,i=1,...,n, be arbitrary complexconstants and[n∏1/2W = (K +iζ)(K − iζ) (K − K i )(K − K i )]. (34.124)i=1Additional quantities K (m) ,m =1,...,n, are defined as solutions of theso-called Jacobi inversion problemn∑K ∫(m)m=1K mwhere the u j are given recursively byK j dKW = u j, j =0,...,n− 1, (34.125)∆u 0 =0, i∂ ζ u j = 1 2 u j−1 + ζ∂ ζ u j−1 ,j=1,...,n. (34.126)All u j are thus solutions of Laplace’s equation. Finally, the Ernst potentialis⎛⎞K⎜n∑∫(m)K n dKE = exp ⎝W− u ⎟n⎠ (34.127)m=1K mThis Ernst potential can also be expressed in terms of theta functions(Neugebauer et al. 1996).For other applications using hyperelliptic functions see Korotkin (1988,1993), Klein and Richter (1997, 1999) and the references given therein.The relation between the different approaches is discussed in Korotkin(1997) and Meinel and Neugebauer (1997).34.7 Other approachesIn this section we shall mention several other approaches to solving Ernst’sequation.