Page 565 - Modelling in Transport Phenomena A Conceptual Approach

B.2. SECONDORDER LINEAR DFFERXNTU EQUATIONS 545

Table B.l Properties of the Bessel functions.

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BEHAVIOR NEAR THE ORIGIN
JJO) = IO(O) = 1
- Yn(0) = Kn(0) = 00 for all n
Jn(0) = In(0) = 0 for n > 0
Note that if the origin is a point in the calculation field, then
Jn(z) and In(z) are the only physically permissible solutions.
BESSEL FUNCTIONS OF NEGATIVE ORDER

J-,(AZ) = (- l)nJ,(Az) Y-,(AX) = (- l)nYn(Az)
Ln(Xx) = In(Az) K-,(XX) = &(AX)

DIFFERENTIAL RELATIONS
d n n
-Jn(XX) = XJn-l(AX) - - &(AX) = - XJ,+1(Xz) + - Jn(Az)
dx X 2
d n n
-Y,(XZ) = AYn-l(Az) - - Yn(Xz) = -AY*+I(Xz) + - Yn(Xs)
dx X X
d n n
--In(A2) = AI,-,(As) - - In(AX) = XIn+l(Az) + - In(Az)
dx 2 2
d n n
-Kn(Az) = - AKn-l(AX) - - &(AX) = - AKn+l(Ax) + - &(AX)
dx X X