In mathematics, the incomplete Bessel functions are types of special functions which act as a type of extension from the complete-type of Bessel functions.
Definition The incomplete Bessel functions are defined as the same delay differential equations of the complete-type Bessel functions:
J v − 1 ( z , w ) − J v + 1 ( z , w ) = 2 ∂ ∂ z J v ( z , w ) {\displaystyle J_{v-1}(z,w)-J_{v+1}(z,w)=2{\dfrac {\partial }{\partial z}}J_{v}(z,w)}
Y v − 1 ( z , w ) − Y v + 1 ( z , w ) = 2 ∂ ∂ z Y v ( z , w ) {\displaystyle Y_{v-1}(z,w)-Y_{v+1}(z,w)=2{\dfrac {\partial }{\partial z}}Y_{v}(z,w)}
I v − 1 ( z , w ) + I v + 1 ( z , w ) = 2 ∂ ∂ z I v ( z , w ) {\displaystyle I_{v-1}(z,w)+I_{v+1}(z,w)=2{\dfrac {\partial }{\partial z}}I_{v}(z,w)}
K v − 1 ( z , w ) + K v + 1 ( z , w ) = − 2 ∂ ∂ z K v ( z , w ) {\displaystyle K_{v-1}(z,w)+K_{v+1}(z,w)=-2{\dfrac {\partial }{\partial z}}K_{v}(z,w)}
H v − 1 ( 1 ) ( z , w ) − H v + 1 ( 1 ) ( z , w ) = 2 ∂ ∂ z H v ( 1 ) ( z , w ) {\displaystyle H_{v-1}^{(1)}(z,w)-H_{v+1}^{(1)}(z,w)=2{\dfrac {\partial }{\partial z}}H_{v}^{(1)}(z,w)}
H v − 1 ( 2 ) ( z , w ) − H v + 1 ( 2 ) ( z , w ) = 2 ∂ ∂ z H v ( 2 ) ( z , w ) {\displaystyle H_{v-1}^{(2)}(z,w)-H_{v+1}^{(2)}(z,w)=2{\dfrac {\partial }{\partial z}}H_{v}^{(2)}(z,w)}
And the following suitable extension forms of delay differential equations from that of the complete-type Bessel functions:
J v − 1 ( z , w ) + J v + 1 ( z , w ) = 2 v z J v ( z , w ) − 2 tanh v w z ∂ ∂ w J v ( z , w ) {\displaystyle J_{v-1}(z,w)+J_{v+1}(z,w)={\dfrac {2v}{z}}J_{v}(z,w)-{\dfrac {2\tanh vw}{z}}{\dfrac {\partial }{\partial w}}J_{v}(z,w)}
… excerpt ends here. Continue reading the full article.
