BESSELS FUNCTION - r[3—7 PROPERTIES OF BESSEL FUNCTIONS...

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Unformatted text preview: r) [3—7] PROPERTIES OF BESSEL FUNCTIONS 109 equation with constant coefficients by the transformation x = e“. The result is f d ' Key, 01% + 72y = 0. (3—131) The general solution of Eq. (3—131), y = e”, readily gives that of Eq. (3—130) in the form ‘ y(x) = (6‘)? = 93- (3—132) Inserting Eq. (3—132) into Eq. (3—130), we obtain the characteristic equation r2 —l— (a — 1)r + 72 = 0. (3—133) Introducing the roots of Eq. (3—133) into Eq. (3—132) yields two particular solutions of Eq. (3—130). For convenience in the solution of problems related to extended surfaces with variable cross sections, the particular solutions of Eq. (3—126) are sum- marized in Table 3—1. 3—7. Properties of Bessel Functions In the properties considered below, 2,, denotes any Bessel function of order 1/, and a: a complex number unless otherwise specified. 1. Bessel functions of the third kind, or H anhel functions of the first and second kinds, of order V are defined to be H§1>'<2>(x) = J,(a;) :l: mm). (3—134) 2. Derivatives of Bessel functions: i meZy_1(m:c), Z = J, Y, I, H”), H”) —rn:cVZ,,_1(7n33), Z = K (3—135) d % [937$an = { i [so—”Z,(inm)] :- {—mx_”Z,+1(7ncc), Z = J, Y,K,H(1),I-I(2) d9: mx_”Z,+1(m33), , Z = I. (3—136) A special case of Eq. (3—136) corresponding to V = O is fl— [zomo] = {—mzl(mx), z = J, Y, K, H“), H”) d-r 7nZ1(m:c), Z = I (3-137) mz,_1(mx) — (V/$)Zy(ma:), Z = J, Y, I, H“), H”) -—7nZ,,_1(mr) —— (V/as)Z,,(in:c), Z = K (3*138) 3%; [24mm = { d —[Z< ~)]— ‘mZvHWHW/xfldm), Z=J,Y,K,H‘“,H(2.) d3: ”mg“ “I mZ,+1(m:c) + (V/x)Z,,(7n3;), Z = I. (3-139) w \0/ /I\ II\ B 4 4. an on 3 0) m G I F F 5 S M E L B \/ x/ \) 4 o m h m m Lu I1 12 L A N O I S N H E M I m E N O Y D A E T S w m m 8 [3—7] PROPERTIES OF BESSEL FUNCTIONS 143 FIG. 3—24 (b) . ’ 1 2 x 3 4 a FIG. 3—24 (d) CONDUCTION HEAT TRANSFER by Vedat S. Arpacz University of Michigan A V‘V ADDISON-WESLEY PUBLISHING COMPANY Reading, Massachusetts Menlo Park, California - London - Amsterdam - Don Mills, Ontario - Sydney ...
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