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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$an = { i [so—”Z,(inm)] : {—mx_”Z,+1(7ncc), Z = J, Y,K,H(1),II(2)
d9: mx_”Z,+1(m33), , Z = I. (3—136)
A special case of Eq. (3—136) corresponding to V = O is fl— [zomo] = {—mzl(mx), z = J, Y, K, H“), H”)
dr 7nZ1(m:c), Z = I (3137) mz,_1(mx) — (V/$)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. (3139) 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 ADDISONWESLEY PUBLISHING COMPANY Reading, Massachusetts
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 Fall '08
 Staff
 Heat Transfer

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