BESSELS FUNCTION

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

This preview shows pages 1–5. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern