Lec4 SpFn Bessel
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Transcript of Lec4 SpFn Bessel
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Special Functions
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Bessel Functions
Applications
EM waves in a cylindrical waveguide
Vibration in a circular membrane
G
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Bessel Functions
Applications
Heat conduction in circular Rods
Frequency Spectrum in FM
G
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Bessel Functions
Bessels Eq. of order , 0 0 is a regular singular point
=
+
=
1 +
=
+
=
++
=
+ 0
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Bessel Functions
=
1 +
=
+
=
+
=
+ 0
1 +
=
1 +
, 0,
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Bessel Functions
0 2 , 2,3, 5 0 1
1
2! 1 2 1=
1! 1
2
+
=
1
! 1
2
+Bessel Functions of the first
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Bessel Functions
=
1! 1 2 +
1 1! 2
2! 2
63! 26
2
21!2!
5252!3!
7273!4!
Bessel Function of the first
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Bessel Functions
Bessels Eq.
0If is not integer, then the general solution is:
= 1! 1 2
+
=
1! 1
2
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Bessel Functions
Ex. Show that the general solution of Bessels Eq.
14 0is:
/ /
where
/ 2 , /
2
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Bessel Functions
1
4 1
2So the solution is given by: / /
/ =
1! 1/2 1
2
/
2=
1! 1/2
2
2
=
1
(2)!
1/2
2
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Bessel FunctionsEx. Find the general solution of Bessels Eq.
0 0,
=
2 =
=
+ =
1 0 0,
14
12! 1 !
2
12! 1
12
2
13! 1
12
13
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Bessel Functions
Properties
For 0,1,2,1. 12. 1
=
1
! 12
=
1+ ! 1
2
+
=
1!
1=
1! 1
2
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Bessel Functions
Bessel Functions of the second kind (Neumann-Weber)
0then the general solution can be written as:
= 1! 1 2
+
lim ()
Heinrich M. Weber
German Math.
1842-1913
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Bessel Functions
Bessel Functions of the second kind (Neumann-Weber)
0then the general solution can be written as:
=
1
! 12
+
lim ()
2
2
2
=
1 1
!
1
2
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Bessel FunctionsEx. Find the general solution of Bessels Eq.
9 4 0 3 3 , 9
4 0
3 3In general, solution of 0
is
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Bessel Functions
Properties
For 0,1,2,1.
2. +
3. + 4. + 2
=
1!
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Bessel Functions
Ex. Express5/ in terms of trigonometric functions+ 2 5/
3/ /
1
Ex. Show that
+
31/ / /
