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Unformatted text preview: 78 Chapter 4 Partial Diﬀerential Equations in Polar and Cylindrical Coordinates On the left, we have the desired integral times (b2 − a2 ) and, on the right, we have
c Jp (bc)aJp (ac) − bJp (ac)Jp (bc) − c aJp (0)Jp (0) − bJp (0)Jp (0) .
Since Jp (0) = 0 if p > 0 and J0(x) = −J1 (x), it follows that Jp (0)Jp (0) −
Jp (0)Jp (0) = 0 for all p > 0. Hence the integral is equal to
c Jp (ax) Jp (bx)x dx = I=
aJp (bc)Jp (ac) − bJp (ac)Jp (bc) .
b2 − a2 Now using the formula
Jp ( x ) = 1
Jp−1(x) − Jp+1 (x) ,
2 we obtain
aJp (bc) Jp−1 (ac) − Jp+1 (ac) − bJp (ac) Jp−1 (bc) − Jp+1 (bc) .
2(b2 − a2) Simplify with the help of the formula
Jp+1 (x) = 2p
Jp (x) − Jp−1 (x)
x and you get
I = = c
aJp (bc) Jp−1 (ac) − ( Jp (ac) − Jp−1(ac))
2(b2 − a2)
−bJp (ac) Jp−1 (bc) − ( Jp (bc) − Jp−1 (bc))
aJp (bc)Jp−1(ac) − bJp (ac)Jp−1 (bc) ,
b2 − a2 as claimed.
Note that this formula implies the orthogonality of Bessel functions. In fact its
proof mirrors the proof of orthogonality from Section 4.8. ...
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This note was uploaded on 12/22/2011 for the course MAP 3305 taught by Professor Stuartchalk during the Fall '11 term at UNF.
- Fall '11