fungsi bessel
TRANSCRIPT
10.4 Buktikanlah (a) J1 /2 (x )=√ 2πx
sin x , (b) J−1 /2 ( x )=√ 2πx
cos x
Pembahasan :
(a)Persamaan umumnya :
Jn ( x )=∑r=0
∞ (−1 )r(x /2)n+2 r
r ! Γ (n+r+1)
J1 /2 (x ) = ∑r=0
∞ (−1 )r( x /2)1/2+2 r
r ! Γ (12+r+1)
= ∑r=0
∞ (−1 )r( x /2)1/2+2 r
r ! Γ (r+3/2)
= (−1 )0(x /2)1 /2+2(0)
0! Γ (0+3 /2)+
(−1 )1(x /2)1/2+2(1)
1 ! Γ (1+3 /2)+
(−1 )2(x /2)1/2+2(2)
2 ! Γ (2+3 /2)+…
= (x /2)1/2
Γ (3/2)−
(x /2)5/2
Γ (5 /2)+
(x /2)9 /2
4 Γ (9 /2)+…
= (x /2)1/2
(1/2)√π−
(x /2)5/2
(3 /2)(1/2)√ π+
(x /2)9 /2
4 (5/2 )(3/2)(1/2)√ π−…
= (x /2)1/2
(1/2)√π (1−(x /2)2
(3/4)+
(x /2)4
(15 /2)−…)
= (x /2)1/2
(1/2)√π (1− x2
22
(3 /4)+
x4
24
(15/2)−…)
= (x /2)1/2
(1/2)√π (1− x2
3+(x /2)3
120−…)
= (x /2)1/2
(1/2)√π (1− x2
3 !+
(x /2)3
5 !−…)
= (x /2)1/2
(1/2)√πsin xx
= √ 2πx
sin x
(b)J−1 /2 ( x ) = ∑r=0
∞ (−1 )r( x /2)−1 /2+2r
r ! Γ (−12
+r+1)
= ∑r=0
∞ (−1 )r( x2 )−1 /2+2 r
r ! Γ (r+1 /2)
= (−1 )0(x /2)−1/2+2(0)
0 ! Γ (0+1/2)+
(−1 )1(x /2)−1 /2+2 (1 )
1 ! Γ (1+1/2)+
(−1 )2(x /2)−1 /2+2(2)
2 ! Γ (2+1/2)+…
= (x /2)−1/2
Γ (1/2)−
(x /2)3 /2
Γ (3 /2)+
(x /2)7/2
4 Γ (5 /2)+…
= (x /2)−1/2
√π−
(x /2)3 /2
(1/2)√π+
(x /2)7 /2
4 (3/2)(1/2)√π−…
= (x /2)−1/2
√π (1−(x /2)2
1/2+
(x /2)4
3−…)
= (x /2)−1/2
√π (1−
x2
22
1 /2+
x4
24
3−…)
= (x /2)−1/2
√π (1− x2
2+
(x /2)3
48−…)
= (x /2)−1/2
√π (1− x2
2 !+(x /2)3
4 !−…)
= (x /2)−1/2
√πcos xx
= √ 2πx
cos x
10.7. Tunjukkanlah bahwa (a) J1 /2 (x )=√ 2πx ( sin x−x cos x
x )
(b) J−1 /2 ( x )=−√ 2πx ( x sin x−cos x
x )Pembahasan :
Dari Soal 10.6 (b) didapatkan persamaan :
Jn−1 (x )+Jn+1 ( x )=2nxJ n( x)
Dan dari soal 10.4. diatas diketahui bahwa
J1 /2 (x ) = √ 2πx
sin x ; J−1 /2 ( x ) = √ 2πx
cos x
dengan mengambil n = ½ maka diperoleh
(a) Jn+1 ( x ) = 2nxJ n ( x )−J n−1(x)
J1 /2+1 ( x ) = 2(1/2)x
J 1/2 ( x )−J1 /2−1(x )
J3 /2 ( x ) = 1xJ 1/2 ( x )−J−1/2( x)
= 1x √ 2πx
sin x−√ 2πx
cos x
= √ 2πx (sin x
x−cos x )
= √ 2πx (sin x−x cos x
x )
(b) dengan mengambil n = - ½ maka diperoleh
Jn−1 (x ) = 2nxJ n ( x )−J n+1(x )
J−1 /2−1 ( x ) = 2(−1/2)x
J−1 /2 (x )−J−1/2+1(x )
J3 /2 ( x ) = −1xJ−1 /2 ( x )−J 1/2(x)
= −1x √ 2
πxcos x−√ 2
πxsin x
= −√ 2πx ( cos x
x−sin x )
= −√ 2πx ( cos x−x sin x
x )