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Additional info for Collisional Transport in Magnetized Plasmas
83) 0 Inside the contour γ+ , the function f (α) satisfies the inequality | f (α)| < M1 exp [(1 − a) Im α] , Im α → +∞. 81). 2. 22]. Let a, b, c, and d be positive numbers; let 0 < b < π /2 and |arg(−ik)| < π /2. We consider a function of r , p(r ) = O(r a−1/2 exp(ikr cos b)), analytic for positive value of r and also in the entire region c < |kr | < ∞, |arg(r )| < 1 . Let k Ψ (α) = −2iπ +∞ p(r ) ikr cos α √ dr. 85) From the theory of the SM transform, Ψ (α) is an analytic function in the domain Re(−ik0 (cos α + cos b)) > 0, as |arg(−ik0 )| < |arg(−ik)| + 1 , and then in the strip |Re(α)| < π /2 − |arg(−ik)| + 1 .
32]) u(kr, ϕ) ∼ n eikr +iπ /4 u n + S(ϕ, ϕ0 ) √ + u+ + u−. 1. 117), res|z=±(3π/2+2Φ) ψΦ (z) = ψΦ (3π/2 − 2Φ) . 154) 1 Remark: Generally, the poles ±(π + Φ + θ± ) − ϕ never coincide with the saddle points ±π since |ϕ| < Φ, and the inequalities 0 < Re θ± < π /2, Im θ± < 0 are assumed. At the same time, the surface waves can noticeably contribute to the total far field only for moderate values of kr . For such kr the numerical values of Φ ∓ ϕ + θ± become relevant. 153) is used for accurate computations of the surface waves.
170); that is, s(z) is a piecewise regular, 4Φ-periodic function. 170). 170) one has i 8Φ iR dτ = cos μτ − sin μz n+ 3 z − 4 4Φ 1 . ) We study the analytic continuation of s(z) from any strip of regularity onto the whole complex plane. It is sufficient to consider the strip −3Φ < Re z < Φ. 171), one can write s(z) = i 8Φ iR F(τ ) − F(z − Φ) z +Φ dτ − F(z − Φ). 172) is a meromorphic function on the whole z-plane. 172) can have singularities at the poles of F(z − Φ) but has no singularities on the line Re z = Φ, that is, the points z = Φ + τ are regular points.