6.2.14 Expansion of Sine and Cosine in Terms of Bessel
Functions
Example 6.2.8 Show that
(i)
024
cos22xJJJ=−+−
(ii)
135
sin2().xJJJ=−+−
Solution We know that
()
1
2
1
0
1
0
1
2
012
2
0
1
()()()()
()[()()]
11
()()()
()((1))()
x
t
t
nnn
nnn
n
nn
nn
n
nnn
n
n
etJxtJxJxtJx
JxtJxtJx
JxtJxtJx
tt
JxttJx
∞−∞
−
=−∞−∞
∞
−
−
=
∞
−
=
==++
=++
⎛⎞⎛⎞
=+−+++
⎜⎟⎜⎟
⎝⎠⎝⎠
=++−
∑∑∑
∑
∑
Put
i
te
θ
=
()222
012
sin
012
()()
(2sin)(2cos2)
ii
xeeiiii
ix
eJeeJeeJ
eJiJJ
θθ
θθθθ
θ
θθ
−
−−−
=+− +++
⇒=+++
Separating the real and imaginary parts we have
024
135
Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month, and much more.