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Bessel Functions

# Bessel Functions - THE FIBER FORUM Fiber Optic...

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THE FIBER FORUM Fiber Optic Communications Dr. JOSEPH C. PALAIS PRESENTED BY

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Dr. Joseph C. Palais Bessel Functions 2 Bessel Functions
Dr. Joseph C. Palais Bessel Functions 3 Bessel’s Equation (1) x 2 (d 2 y/dx 2 ) + x(dy/dx) + (x 2 - l 2 )y =0 Let l = an integer. There are two solutions. (2) ( 29 ( 29 ( 29 + = - � � = = � � + + � � ° r 2r r 0 1 x y J x r!Γ r 1 2 l l l This is a Bessel function of the first kind with order l .

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Dr. Joseph C. Palais Bessel Functions 4 ( 29 ( 29 ( 29 - - + + = + = Γ Γ Γ n 1 t 0 r 1 n t e dt n 1 n! l Gamma function: n > 0 n Γ (n) 0 Γ (1) = 0! = 1 1 Γ (2) = 1! = 1 2 Γ (3) = 2! = 2 3 Γ (4) = 3! = 6
Dr. Joseph C. Palais Bessel Functions 5 ( 29 ( ) - H = = sin J (x) cos( ) - J (x) y Y x lim l l ν ν υπ υπ ν This is a Bessel function of the second kind of order l . Y l (x) is the Neumann’s function (also known as the Weber function) (3)

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Dr. Joseph C. Palais Bessel Functions 6 For x small (approaching zero), the following approximations are useful: (4) J l (x) = [1/ Γ ( l + 1)] (x/2) l    for all l (5) Y 0 (x) = [2/ π ] ln (x) (6) Y l (x) = [- Γ ( l ) / π ] (2/x) l   l > 0
Dr. Joseph C. Palais Bessel Functions 7

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Bessel Functions - THE FIBER FORUM Fiber Optic...

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