Definition. For x > 0 we define the Gamma function ( x ) by the - PDF document

Definition. For x > 0 we define the Gamma function Γ( x ) by the formula ∞ � t x − 1 e − t dt. Γ( x ) = 0 1

Theorem. The Gamma function satisfies Γ( x + 1) = x Γ( x ) for all x ≥ 0. 2

Theorem. (i) Γ( x ) = ( x − 1)! for all x ∈ I N . = √ π . � 1 � (ii) Γ 2 3

Theorem. 1 � � Let m ∈ I N 0 . Then lim converges. Γ( z ) z →− m 1 1 � � If we define the function g ( z ) as Γ( z ) for − z / ∈ I N 0 and g ( z ) = lim for − z ∈ I N 0 , Γ( w ) w → z then g is an entire function on the complex plane. 4

Definition. By a Bessel equation of order m we mean the equation x 2 y ′′ + xy ′ + ( x 2 − m 2 ) y = 0 , x > 0 . 5

Definition. Let m ∈ I R satisfy − m / ∈ I N . We define the Bessel function (of the first kind) of order m by the formula ∞ ( − 1) k � x � 2 k + m � J m ( x ) = , x > 0 . k !Γ( m + k + 1) 2 k =0 � � For m ∈ I N we define J − m ( x ) = lim J µ ( x ) . µ →− m 6

Theorem. For all m the function J m solves the Bessel equation of order | m | . If m is not an integer, then the set { J m , J − m } is linearly independent and hence a basis of solutions of the Bessel equation of order m . 7

Theorem. If m is an integer, then J − m ( x ) = ( − 1) m J m ( x ) for all x > 0. 8

Theorem. � � 2 2 (i) J 1 / 2 ( x ) = πx sin( x ) and J − 1 / 2 ( x ) = πx cos( x ) for x ≥ 0. (ii) If x is small and positive, 0 < x < √ x + 1, then 1 � x � m . J m ( x ) ∼ 2 Γ( m + 1) (iii) If x is large, then � 2 x − π � � J m ( x ) ∼ πx cos 4 (2 m + 1) . 9

Definition. We define Bessel functions of the second kind of order m as follows: If m is not an integer, we set N m ( x ) = J m ( x ) cos( mπ ) − J − m ( x ) . sin( mπ ) If m is an integer, we set N m ( x ) = lim µ → m ( N µ ( x )) for x > 0. 10

Theorem. For all m ∈ I R , the set { J m , N m } is a fundamental system of the Bessel equation of order m . 11

Theorem. Let m ≥ 0. (i) If x is small and positive, 0 < x < √ x + 1, then N 0 ( x ) ∼ 2 � x � � � ln + 0 . 5772 ... and π 2 � 2 N m ( x ) ∼ − Γ( m ) � m pro m > 0 . π x (ii) If x is large, then � 2 x − π � � N m ( x ) ∼ πx sin 4 (2 m + 1) . 12