Geology Reference
In-Depth Information
Tabl e B. 1 Sample values of Legendre functions of the first kind calculated with
the programme LEGENDRE.FOR.
P 1 ( z )
P 2 ( z )
P 2 ( z )
P 2 ( z )
P 3 ( z )
P 3 ( z )
P 3 ( z )
P 3 ( z )
z
1.0
0.0000
1.0000
0.0000
0.0000
1.0000
0.0000
0.0000
0.0000
0.9
0.4359
0.7150
1.1769
0.5700
0.4725
1.9942
2.5650
1.2423
0.8
0.6000
0.4600
1.4400
1.0800
0.0800
1.9800
4.3200
3.2400
0.7
0.7141
0.2350
1.4997
1.5300
0.1925
1.5533
5.3550
5.4632
0.6
0.8000
0.0400
1.4400
1.9200
0.3600
0.9600
5.7600
7.6800
0.5
0.8660
0.1250
1.2990
2.2500
0.4375
0.3248
5.6250
9.7428
0.4
0.9165
0.2600
1.0998
2.5200
0.4400
0.2750
5.0400
11.5481
0.3
0.9539
0.3650
0.8585
2.7300
0.3825
0.7870
4.0950
13.0213
0.2
0.9798
0.4400
0.5879
2.8800
0.2800
1.1758
2.8800
14.1091
0.1
0.9950
0.4850
0.2985
2.9700
0.1475
1.4179
1.4850
14.7756
0.0
1.0000
0.5000
0.0000
3.0000
0.0000
1.5000
0.0000
15.0000
P3=((2.D0*AI-1.D0)*Z*P2-(AI+AM-1.D0)*P1)/(AI-AM)
P1=P2
P2=P3
16
CONTINUE
PMN=P3
GO TO 11
14
PMN=PMM
GO TO 11
15
PMN=PMMP1
11
CONTINUE
RETURN
END
Sample values of P n ( z ) as output of these codes are given in Table B.1.
The first few Legendre functions in explicit form, beginning with the Legendre
polynomials, are as follows:
2 3 z 2
1 ,
2 5 z 3
3 z ,
1
1
P 0 ( z )
=
1,
P 2 ( z )
=
P 3 ( z )
=
P 2 ( z ) =− 3 1 z 2 1/2 z , P 3 ( z ) =−
2 1 z 2 1/2 5 z 2
1 ,
3
P 1 ( z ) = z ,
3 1
z 2 ,
15 1
z 2 z ,
P 1 ( z )
z ) 1/2
, P 2 ( z )
P 3 ( z )
=−
(1
=
=
15 1
z 2 3/2
P 3 ( z )
=−
.
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