The symmetrized Fermi function and its transforms

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6530 D W L Sprung and J Martorell and α = λπ = πqd (in this section, z is real). Using now D = d/dz, equation (15) becomes q 2 v(q) = αD H(z). (30) sin αD Expanding in powers of α = πqd, and taking the derivatives, one finds for the first few terms q 2 v(q) = qRJ 1 (qR) − (πqd)2 [qRJ 1 (qR) − J 0 (qR)] 6 [ + 7(πqd)4 qRJ 1 (qR) − 2J 0 (qR) + J ] 1(qR) 360 qR − 31(πqd)6 360(42) [ qRJ 1 (qR) − 3J 0 (qR) + 3 qR J 1(qR) − 3 (qR) 2 J 2(qR) ] +···. (31) As noted in appendix A, this series diverges for large |α|, so one would like to know its sum. Further study reveals that the coefficients involve functions y p (z) ≡ (−) p+1 (2p − 1)!! J p(z) (32) z p with y 0 =−J 0 (z) and y −1 ≡ zJ 1 (z). Differentiation is facilitated by the relation y p ′′ =−y p−y p+1 . (33) For example, one has (zJ 1 (z)) ′′ =−zJ 1 (z) + J 0 =−y −1 −y 0 . (34) At large z, the J 0 term is of relative order 1/z. If we could neglect it altogether, then we could replace the derivative operator in equation (30) by i since only even powers of D occur in the expansion. The result would be zJ 1 (z)α/ sinh α, as in equation (27). Taking the derivatives systematically leads to the tableau −(zJ 1 ) (2) = zJ 1 + y 0 (zJ 1 ) (4) = zJ 1 + 2y 0 + y 1 −(zJ 1 ) (6) = zJ 1 + 3y 0 + 3y 1 + y 2 (35) (zJ 1 ) (8) = zJ 1 + 4y 0 + 6y 1 + 4y 2 + y 3 −(zJ 1 ) (10) = zJ 1 + 5y 0 + 10y 1 + 10y 2 + 5y 3 + y 4 . One sees that the rule for carrying out the derivatives generates coefficients according to Pascal’s triangle, so they are the binomial coefficients l C p . The general case is therefore l∑ (−) l D 2l zJ 1 (z) = lC p y p−1 . (36) p=0 Following equation (A.3), with c 0 = 1 , we have 2 αD ∞∑ sinh αD = 2 (−) l c 2l α 2l D 2l . (37) l=0 This gives ∞∑ l∑ q 2 v(q) = 2 (−) l c 2l α 2l lC p y p−1 . (38) l=0 p=0
Symmetrized Fermi function 6531 By reversing the order of summations we have ∞∑ ∞∑ q 2 v(q) = 2 (−) l lC p c 2l α 2l . (39) p=1 y p−1 p=0 l=p The coefficient of y p−1 can be expressed in closed form if we note from equation (37) (with D = 1) that α 2p ( ) ∂ p α ∞∑ p! ∂α 2 sinh α = (−) l c 2ll C p α 2l . (40) l=p Thus, q 2 α ∞∑ v(q) − zJ 1 (z) sinh α = 1 ∂ p α y p−1 p! α2p ∂(α p=1 2 ) p sinh α ∞∑ ( ) 1 1 ∂ p α = y p−1 2 p p! α2p α ∂α sinh α ) ∂ p α ∂α sinh α . (41) ∞∑ ( p (2p − 3)!! J p−1 (z) 1 = (−) α 2p 2 p=1 p p! z p−1 α We worked out the first few terms of this series explicitly, with the result q 2 v(q) = πqd [ qRJ 1 (qR) + 1 ( ) πqd sinh πqd 2 tanh πqd −1 J 0 (qR) + 1 [( )( ) ] 2πqd πqd 8 tanh πqd +1 tanh πqd −1 −(πqd) 2 J1 (qR) qR + 1 [ ( 2(πqd) 2 3 16 tanh 2 πqd + 2πqd )( ) πqd tanh πqd +1 tanh πqd −1 − 5(πqd)3 tanh πqd × J 2(qR) (qR) 2 +···] . (42) It is seen that α/ sinh α is a common factor, and the remaining coefficients become polynomials in α at large qd, soq 2 v(q) decreases exponentially. The coefficients of y p become more and more complicated but one can certainly work them out systematically. The result is an asymptotic series in z, where the dependence on α has been summed. One is not limited to small qd, as for the Sommerfeld expansion, or equation (31). On the other hand, if equation (42) is expanded in powers of α, it reproduces exactly equation (31). In figure 1 we show a log plot of |v(q)| versus q for the case R = 30, d = 1. The maxima of |v(q)| vary by 20 orders of magnitude over the range considered, and on this scale the exact (numerical) values cannot be distinguished from expression (42). In figure 2, we have expanded a small portion of the drawing, for 2.2
Page 1 and 2: J. Phys. A: Math. Gen. 30 (1997) 65
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