Higher Engineering Mathematics, Sixth Edition

510 Higher Engineering Mathematics

i.e. y=AJv(x)+BJ−v(x)

=A

(x

2

)v{ 1

(v+1)

−

x^2

22 (1!)(v+2)

+

x^4

24 (2!)(v+4)

−···

}

+B

(x

2

)−v{ 1

(1−v)

−

x^2

22 (1!)(2−v)

+

x^4

24 (2!)(3−v)

−···

}

In general terms:Jv(x)=

(x

2

)v∑∞

k= 0

(− 1 )kx^2 k

22 k(k!)(v+k+ 1 )

andJ−v(x)=

(x

2

)−v∑∞

k= 0

(− 1 )kx^2 k

22 k(k!)(k−v+ 1 )

Another Bessel function

It may be shown that another series forJn(x)is given by:

Jn(x)=

(x

2

)n{ 1

n!

−

1

(n+ 1 )!

(x

2

) 2

+

1

(2!)(n+ 2 )!

(x

2

) 4

− ···

}

From this series two commonly used function are

derived,

i.e. J 0 (x)=

1

(0!)

−

1

(1!)^2

(x

2

) 2

+

1

(2!)^2

(x

2

) 4

−

1

(3!)^2

(x

2

) 6

+···

= 1 −

x^2

22 (1!)^2

+

x^4

24 (2!)^2

−

x^6

26 (3!)^2

+···

and J 1 (x)=

x

2

{

1

(1!)

−

1

(1!)(2!)

(x

2

) 2

+

1

(2!)(3!)

(x

2

) 4

−···

}

=

x

2

−

x^3

23 (1!)(2!)

+

x^5

25 (2!)(3!)

−

x^7

27 (3!)(4!)

+···

Tables of Bessel functions are available for a range of

values ofnandx, and inthese,J 0 (x)andJ 1 (x)are most

commonly used.

Graphs ofJ 0 (x), which looks similar to a cosine, and

J 1 (x), which looks similar to a sine, are shown in

Figure 52.1.

yJ 0 (x)

yJ 1 (x)

1

0.5

0.5

(^024681214) x
y
10
Figure 52.1