6� > DrawBasic:Resources.Calculus.SpecialFns.Bessel

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$� copyright Joe Taylor June 1994
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;� N.B. For ERROR THROWBACK to work The first line above
7�      MUST have the correct pathname of this file.
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H� ------------------------------------------------------------------
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� References
� ==========
�
>�  1. 'Handbook of Mathematical Functions' Dover Chapter 9
$�      M.Abramowitz & I.A.Stegun
�
.�  2. 'Numerical Recipes' CUP,  pp.170-180
B�      W.H.Press, B.P.Flannery, S.A.Teukolsky & W.T.Vetterling
�
�
�  N.B.
�
F�      Bessel functions are solutions of the differential equation
�
<�             z^2 (d/dz)^2 w + z dw/dz + (z^2-n^2) w = 0
 �
!�
"G� -----------------------------------------------------------------
#
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%*� ------------------------------------
&�
')� 1. Bessel functions of the 1st kind
(�
)*� ------------------------------------
*
+� �BesselJ0(x)
,� i%,y,result,ax,y,z,xx
-� �
.� �x < 8 �
/
 y=x*x
0	 � +0
1Q � -184.9052456,77392.33017,-11214424.18,651619640.7,-13362590354,57568490574
2C � 1,267.8532712,59272.64853,9494680.718,1029532985,57568490411
3- result=�Polynomial(y,6)/�Polynomial(y,6)
4�
5
 ax=�x
6 z=8/ax
7
 y=z*z
8 xx=ax-�/4
9	 � +0
:G � .2093887211E-6,-.20733706397E-5,.2734510407E-4,-.1098628627E-2,1
;4 � -.934945152E-7,.7621095161E-6,-.6911147651E-5
<% � .1430488765E-3,-.1562499995E-1
=; result=�(xx)*�Polynomial(y,5)-z*�(xx)*�Polynomial(y,5)
> result=�(2/ax/�)*result
?�
@� �
A=result
B
C� �BesselJ1(x)
D� i%,y,result,ax,y,z,xx
E� �
F� �x < 8 �
G
 y=x*x
H	 � +0
IO � -30.16036606,15704.4826,-2972611.439,242396853.1,-7895059235,72362614232
JD � 1,376.9991397,99447.43394,18583304.74,2300535178,144725228442
K/ result=x*�Polynomial(y,6)/�Polynomial(y,6)
L�
M
 ax=�x
N z=8/ax
O
 y=z*z
P xx=ax-3*�/4
Q	 � +0
RA � -.240337019E-6,.2457520174E-5,-.3516396496E-4,.183105E-2,1
SN � .105787412E-6,-.88228987E-6,.8449199096E-5,-.2002690873E-3,.04687499995
TB result=(�(xx)*�Polynomial(y,5)-z*�(xx)*�Polynomial(y,5))*�(x)
U result=�(2/ax/�)*result
V�
W� �
X=result
Y
Z� �BesselJ(x,order%)
[
� B,ax
\Ȏ order% �
] � 0 : B=�BesselJ0(x)
^ � 1 : B=�BesselJ1(x)
_ 
`  ax=�x
a  Ȏ � �
b   � ax=0      : B=0
c:   � ax>order% : B=�BesselForwardRecurrence(ax,order%)
d6         : B=�BesselBackwardRecurrence(ax,order%)
e  �
f�
g=B
h
i)� �BesselForwardRecurrence(ax,order%)
j� B0,B1,B,jj%,tox
ktox=2/ax
l'B0=�BesselJ0(ax) : B1=�BesselJ1(ax)
m� jj%=1 � order%-1
n B=jj%*tox*B1-B0
o B0=B1 : B1=B
p �
q=B
r
s*� �BesselBackwardRecurrence(ax,order%)
t(� BJP,BJ,BJM,jj%,tox,sum,bessj,jsum%
utox=2/ax
v(mm%=2*((order%+�(�(order%*40))) � 2)
w,BJP=0 : BJ=1 : sum=0 : jsum%=0 : bessj=0
x� jj%=mm% � 1 �-1
y BJM=jj%*tox*BJ-BJP
z BJP=BJ : BJ=BJM
{ � �(BJ)>1E10 �
|A  BJ=BJ/1E10 : BJP=BJP/1E10 : bessj=bessj/1E10 : sum=sum/1E10
}  �
~ � jsum%<>0 � sum+=BJ
 jsum%=1-jsum%
� � jj%=order% � bessj=BJP
� �
� sum=2*sum-BJ
� =bessj/sum
�
�8� �BesselJ_Asymptotic(x,n%)=�(2/�/x)*�(x-�*n%/2-�/4)
�
�*� ------------------------------------
��
�*� 2. Bessel functions of the 2cnd kind
��
�*� ------------------------------------
�
�� �BesselY0(x)
� � Not defined for negative x
�� i%,y,result,y,z,xx
�� �
�
� x < 8 �
�
 y=x*x
�	 � +0
�O � 228.4622733,-86327.92757,10879881.29,-512359803.6,7062834065,-2957821389
�C � 1,226.1030244,47447.2647,7189466.438,745249964.8,40076544269
�J result=�Polynomial(y,6)/�Polynomial(y,6)+.636619772*�BesselJ0(x)*�(x)
��
�
 z=8/x
�
 y=z*z
� xx=x-.785398164
�	 � +0
�F � .2093887211E-6,-.2073370639E-5,.2734510407E-4,-.1098628627E-2,1
�C � -.934945152E-7,.7621095161E-6,-.6911147651E-5,.1430488765E-3
� � -.1562499995E-1
� result=�(.636619772/x)
�D result=result*(�(xx)*�Polynomial(y,5)+z*�(xx)*�Polynomial(y,5))
��
�� �
�=result
�
�� �BesselY1(x)
� � Not defined for negative x
�� i%,y,result,y,z,xx
�� �
�
� x < 8 �
�
 y=x*x
�	 � +0
�A � .8511937935E4,-.4237922726E7,.7349264551E9,-.5153438139E11
�% � .1275274390E13,-.4900604943E13
� result=x*�Polynomial(y,6)
�@ � 1,.3549632885E3,.102042605E6,.2245904002E8,.3733650367E10
�# � .4244419664E12,.249958057E14
�F result=result/�Polynomial(y,7)+.636619772*(�BesselJ1(x)*�(x)-1/x)
��
�
 z=8/x
�
 y=z*z
� xx=x-2.356194491
�	 � +0
�A � -.240337019E-6,.2457520174E-5,-.3516396496E-4,.183105E-2,1
�" result=�(xx)*�Polynomial(y,5)
�N � .105787412E-6,-.88228987E-6,.8449199096E-5,-.2002690873E-3,.04687499995
�% result+=z*�(xx)*�Polynomial(y,5)
�" result=�(.636619772/x)*result
��
�� �
�=result
�
�� �BesselY(x,order%)
�� B,factor,b1,b0,ii%
�Ȏ order% �
� � 0
�  B=�BesselY0(x)
� � 1
�  B=�BesselY1(x)
� 
�  factor=2/x
�'  b1=�BesselY1(x) : b0=�BesselY0(x)
�  � ii%=1 � order%-1
�   B=ii%*factor*b1-b0
�   b0=b1 : b1=B
�   �
��
�=B
�
�*� ------------------------------------
��
�)� 3. Utility : Evaluating Polynomials
��
�*� ------------------------------------
�
�� �Polynomial(x,n%)
�� i%,coeff,P
�P=0
�� i%=1 � n%
� � coeff
� P=coeff+P*x
� �
�=P
�

00000000  0d 00 01 36 f4 20 3e 20  44 72 61 77 42 61 73 69  |...6. > DrawBasi|
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00000020  75 6c 75 73 2e 53 70 65  63 69 61 6c 46 6e 73 2e  |ulus.SpecialFns.|
00000030  42 65 73 73 65 6c 0d 00  02 04 0d 00 03 3d f4 20  |Bessel.......=. |
00000040  3d 3d 3d 3d 3d 3d 3d 3d  3d 3d 3d 3d 3d 3d 3d 3d  |================|
*
00000070  3d 3d 3d 3d 3d 3d 3d 0d  00 04 05 f4 0d 00 05 24  |=======........$|
00000080  f4 20 63 6f 70 79 72 69  67 68 74 20 4a 6f 65 20  |. copyright Joe |
00000090  54 61 79 6c 6f 72 20 4a  75 6e 65 20 31 39 39 34  |Taylor June 1994|
000000a0  0d 00 06 05 f4 0d 00 07  3b f4 20 4e 2e 42 2e 20  |........;. N.B. |
000000b0  46 6f 72 20 45 52 52 4f  52 20 54 48 52 4f 57 42  |For ERROR THROWB|
000000c0  41 43 4b 20 74 6f 20 77  6f 72 6b 20 54 68 65 20  |ACK to work The |
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000000e0  0d 00 08 37 f4 20 20 20  20 20 20 4d 55 53 54 20  |...7.      MUST |
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*
000001b0  3d 3d 3d 0d 00 0d 05 f4  0d 00 0e 05 f4 0d 00 0f  |===.............|
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*
00000200  2d 2d 2d 2d 2d 0d 00 10  05 f4 0d 00 11 10 f4 20  |-----.......... |
00000210  52 65 66 65 72 65 6e 63  65 73 0d 00 12 10 f4 20  |References..... |
00000220  3d 3d 3d 3d 3d 3d 3d 3d  3d 3d 0d 00 13 05 f4 0d  |==========......|
00000230  00 14 3e f4 20 20 31 2e  20 27 48 61 6e 64 62 6f  |..>.  1. 'Handbo|
00000240  6f 6b 20 6f 66 20 4d 61  74 68 65 6d 61 74 69 63  |ok of Mathematic|
00000250  61 6c 20 46 75 6e 63 74  69 6f 6e 73 27 20 44 6f  |al Functions' Do|
00000260  76 65 72 20 43 68 61 70  74 65 72 20 39 0d 00 15  |ver Chapter 9...|
00000270  24 f4 20 20 20 20 20 20  4d 2e 41 62 72 61 6d 6f  |$.      M.Abramo|
00000280  77 69 74 7a 20 26 20 49  2e 41 2e 53 74 65 67 75  |witz & I.A.Stegu|
00000290  6e 0d 00 16 05 f4 0d 00  17 2e f4 20 20 32 2e 20  |n..........  2. |
000002a0  27 4e 75 6d 65 72 69 63  61 6c 20 52 65 63 69 70  |'Numerical Recip|
000002b0  65 73 27 20 43 55 50 2c  20 20 70 70 2e 31 37 30  |es' CUP,  pp.170|
000002c0  2d 31 38 30 0d 00 18 42  f4 20 20 20 20 20 20 57  |-180...B.      W|
000002d0  2e 48 2e 50 72 65 73 73  2c 20 42 2e 50 2e 46 6c  |.H.Press, B.P.Fl|
000002e0  61 6e 6e 65 72 79 2c 20  53 2e 41 2e 54 65 75 6b  |annery, S.A.Teuk|
000002f0  6f 6c 73 6b 79 20 26 20  57 2e 54 2e 56 65 74 74  |olsky & W.T.Vett|
00000300  65 72 6c 69 6e 67 0d 00  19 05 f4 0d 00 1a 05 f4  |erling..........|
00000310  0d 00 1b 0b f4 20 20 4e  2e 42 2e 0d 00 1c 05 f4  |.....  N.B......|
00000320  0d 00 1d 46 f4 20 20 20  20 20 20 42 65 73 73 65  |...F.      Besse|
00000330  6c 20 66 75 6e 63 74 69  6f 6e 73 20 61 72 65 20  |l functions are |
00000340  73 6f 6c 75 74 69 6f 6e  73 20 6f 66 20 74 68 65  |solutions of the|
00000350  20 64 69 66 66 65 72 65  6e 74 69 61 6c 20 65 71  | differential eq|
00000360  75 61 74 69 6f 6e 0d 00  1e 05 f4 0d 00 1f 3c f4  |uation........<.|
00000370  20 20 20 20 20 20 20 20  20 20 20 20 20 7a 5e 32  |             z^2|
00000380  20 28 64 2f 64 7a 29 5e  32 20 77 20 2b 20 7a 20  | (d/dz)^2 w + z |
00000390  64 77 2f 64 7a 20 2b 20  28 7a 5e 32 2d 6e 5e 32  |dw/dz + (z^2-n^2|
000003a0  29 20 77 20 3d 20 30 0d  00 20 05 f4 0d 00 21 05  |) w = 0.. ....!.|
000003b0  f4 0d 00 22 47 f4 20 2d  2d 2d 2d 2d 2d 2d 2d 2d  |..."G. ---------|
000003c0  2d 2d 2d 2d 2d 2d 2d 2d  2d 2d 2d 2d 2d 2d 2d 2d  |----------------|
*
000003f0  2d 2d 2d 2d 2d 2d 2d 2d  0d 00 23 04 0d 00 24 04  |--------..#...$.|
00000400  0d 00 25 2a f4 20 2d 2d  2d 2d 2d 2d 2d 2d 2d 2d  |..%*. ----------|
00000410  2d 2d 2d 2d 2d 2d 2d 2d  2d 2d 2d 2d 2d 2d 2d 2d  |----------------|
00000420  2d 2d 2d 2d 2d 2d 2d 2d  2d 2d 0d 00 26 05 f4 0d  |----------..&...|
00000430  00 27 29 f4 20 31 2e 20  42 65 73 73 65 6c 20 66  |.'). 1. Bessel f|
00000440  75 6e 63 74 69 6f 6e 73  20 6f 66 20 74 68 65 20  |unctions of the |
00000450  31 73 74 20 6b 69 6e 64  0d 00 28 05 f4 0d 00 29  |1st kind..(....)|
00000460  2a f4 20 2d 2d 2d 2d 2d  2d 2d 2d 2d 2d 2d 2d 2d  |*. -------------|
00000470  2d 2d 2d 2d 2d 2d 2d 2d  2d 2d 2d 2d 2d 2d 2d 2d  |----------------|
00000480  2d 2d 2d 2d 2d 2d 2d 0d  00 2a 04 0d 00 2b 12 dd  |-------..*...+..|
00000490  20 a4 42 65 73 73 65 6c  4a 30 28 78 29 0d 00 2c  | .BesselJ0(x)..,|
000004a0  1b ea 20 69 25 2c 79 2c  72 65 73 75 6c 74 2c 61  |.. i%,y,result,a|
000004b0  78 2c 79 2c 7a 2c 78 78  0d 00 2d 07 ea 20 dc 0d  |x,y,z,xx..-.. ..|
000004c0  00 2e 0e e7 20 94 78 20  3c 20 38 20 8c 0d 00 2f  |.... .x < 8 .../|
000004d0  0a 20 79 3d 78 2a 78 0d  00 30 09 20 f7 20 2b 30  |. y=x*x..0. . +0|
000004e0  0d 00 31 51 20 dc 20 2d  31 38 34 2e 39 30 35 32  |..1Q . -184.9052|
000004f0  34 35 36 2c 37 37 33 39  32 2e 33 33 30 31 37 2c  |456,77392.33017,|
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00000520  39 30 33 35 34 2c 35 37  35 36 38 34 39 30 35 37  |90354,5756849057|
00000530  34 0d 00 32 43 20 dc 20  31 2c 32 36 37 2e 38 35  |4..2C . 1,267.85|
00000540  33 32 37 31 32 2c 35 39  32 37 32 2e 36 34 38 35  |32712,59272.6485|
00000550  33 2c 39 34 39 34 36 38  30 2e 37 31 38 2c 31 30  |3,9494680.718,10|
00000560  32 39 35 33 32 39 38 35  2c 35 37 35 36 38 34 39  |29532985,5756849|
00000570  30 34 31 31 0d 00 33 2d  20 72 65 73 75 6c 74 3d  |0411..3- result=|
00000580  a4 50 6f 6c 79 6e 6f 6d  69 61 6c 28 79 2c 36 29  |.Polynomial(y,6)|
00000590  2f a4 50 6f 6c 79 6e 6f  6d 69 61 6c 28 79 2c 36  |/.Polynomial(y,6|
000005a0  29 0d 00 34 05 cc 0d 00  35 0a 20 61 78 3d 94 78  |)..4....5. ax=.x|
000005b0  0d 00 36 0b 20 7a 3d 38  2f 61 78 0d 00 37 0a 20  |..6. z=8/ax..7. |
000005c0  79 3d 7a 2a 7a 0d 00 38  0e 20 78 78 3d 61 78 2d  |y=z*z..8. xx=ax-|
000005d0  af 2f 34 0d 00 39 09 20  f7 20 2b 30 0d 00 3a 47  |./4..9. . +0..:G|
000005e0  20 dc 20 2e 32 30 39 33  38 38 37 32 31 31 45 2d  | . .2093887211E-|
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00000610  34 2c 2d 2e 31 30 39 38  36 32 38 36 32 37 45 2d  |4,-.1098628627E-|
00000620  32 2c 31 0d 00 3b 34 20  dc 20 2d 2e 39 33 34 39  |2,1..;4 . -.9349|
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00000670  35 36 32 34 39 39 39 39  35 45 2d 31 0d 00 3d 3b  |562499995E-1..=;|
00000680  20 72 65 73 75 6c 74 3d  9b 28 78 78 29 2a a4 50  | result=.(xx)*.P|
00000690  6f 6c 79 6e 6f 6d 69 61  6c 28 79 2c 35 29 2d 7a  |olynomial(y,5)-z|
000006a0  2a b5 28 78 78 29 2a a4  50 6f 6c 79 6e 6f 6d 69  |*.(xx)*.Polynomi|
000006b0  61 6c 28 79 2c 35 29 0d  00 3e 1c 20 72 65 73 75  |al(y,5)..>. resu|
000006c0  6c 74 3d b6 28 32 2f 61  78 2f af 29 2a 72 65 73  |lt=.(2/ax/.)*res|
000006d0  75 6c 74 0d 00 3f 05 cd  0d 00 40 07 f7 20 dc 0d  |ult..?....@.. ..|
000006e0  00 41 0b 3d 72 65 73 75  6c 74 0d 00 42 04 0d 00  |.A.=result..B...|
000006f0  43 12 dd 20 a4 42 65 73  73 65 6c 4a 31 28 78 29  |C.. .BesselJ1(x)|
00000700  0d 00 44 1b ea 20 69 25  2c 79 2c 72 65 73 75 6c  |..D.. i%,y,resul|
00000710  74 2c 61 78 2c 79 2c 7a  2c 78 78 0d 00 45 07 ea  |t,ax,y,z,xx..E..|
00000720  20 dc 0d 00 46 0e e7 20  94 78 20 3c 20 38 20 8c  | ...F.. .x < 8 .|
00000730  0d 00 47 0a 20 79 3d 78  2a 78 0d 00 48 09 20 f7  |..G. y=x*x..H. .|
00000740  20 2b 30 0d 00 49 4f 20  dc 20 2d 33 30 2e 31 36  | +0..IO . -30.16|
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00000780  30 35 39 32 33 35 2c 37  32 33 36 32 36 31 34 32  |059235,723626142|
00000790  33 32 0d 00 4a 44 20 dc  20 31 2c 33 37 36 2e 39  |32..JD . 1,376.9|
000007a0  39 39 31 33 39 37 2c 39  39 34 34 37 2e 34 33 33  |991397,99447.433|
000007b0  39 34 2c 31 38 35 38 33  33 30 34 2e 37 34 2c 32  |94,18583304.74,2|
000007c0  33 30 30 35 33 35 31 37  38 2c 31 34 34 37 32 35  |300535178,144725|
000007d0  32 32 38 34 34 32 0d 00  4b 2f 20 72 65 73 75 6c  |228442..K/ resul|
000007e0  74 3d 78 2a a4 50 6f 6c  79 6e 6f 6d 69 61 6c 28  |t=x*.Polynomial(|
000007f0  79 2c 36 29 2f a4 50 6f  6c 79 6e 6f 6d 69 61 6c  |y,6)/.Polynomial|
00000800  28 79 2c 36 29 0d 00 4c  05 cc 0d 00 4d 0a 20 61  |(y,6)..L....M. a|
00000810  78 3d 94 78 0d 00 4e 0b  20 7a 3d 38 2f 61 78 0d  |x=.x..N. z=8/ax.|
00000820  00 4f 0a 20 79 3d 7a 2a  7a 0d 00 50 10 20 78 78  |.O. y=z*z..P. xx|
00000830  3d 61 78 2d 33 2a af 2f  34 0d 00 51 09 20 f7 20  |=ax-3*./4..Q. . |
00000840  2b 30 0d 00 52 41 20 dc  20 2d 2e 32 34 30 33 33  |+0..RA . -.24033|
00000850  37 30 31 39 45 2d 36 2c  2e 32 34 35 37 35 32 30  |7019E-6,.2457520|
00000860  31 37 34 45 2d 35 2c 2d  2e 33 35 31 36 33 39 36  |174E-5,-.3516396|
00000870  34 39 36 45 2d 34 2c 2e  31 38 33 31 30 35 45 2d  |496E-4,.183105E-|
00000880  32 2c 31 0d 00 53 4e 20  dc 20 2e 31 30 35 37 38  |2,1..SN . .10578|
00000890  37 34 31 32 45 2d 36 2c  2d 2e 38 38 32 32 38 39  |7412E-6,-.882289|
000008a0  38 37 45 2d 36 2c 2e 38  34 34 39 31 39 39 30 39  |87E-6,.844919909|
000008b0  36 45 2d 35 2c 2d 2e 32  30 30 32 36 39 30 38 37  |6E-5,-.200269087|
000008c0  33 45 2d 33 2c 2e 30 34  36 38 37 34 39 39 39 39  |3E-3,.0468749999|
000008d0  35 0d 00 54 42 20 72 65  73 75 6c 74 3d 28 9b 28  |5..TB result=(.(|
000008e0  78 78 29 2a a4 50 6f 6c  79 6e 6f 6d 69 61 6c 28  |xx)*.Polynomial(|
000008f0  79 2c 35 29 2d 7a 2a b5  28 78 78 29 2a a4 50 6f  |y,5)-z*.(xx)*.Po|
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