Incomplete Bessel functions

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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.

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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)} Y v − 1 ( z , w ) + Y v + 1 ( z , w ) = 2 v z Y v ( z , w ) − 2 tanh ⁡ v w z ∂ ∂ w Y v ( z , w ) {\displaystyle Y_{v-1}(z,w)+Y_{v+1}(z,w)={\dfrac {2v}{z}}Y_{v}(z,w)-{\dfrac {2\tanh vw}{z}}{\dfrac {\partial }{\partial w}}Y_{v}(z,w)} I v − 1 ( z , w ) − I v + 1 ( z , w ) = 2 v z I v ( z , w ) − 2 tanh ⁡ v w z ∂ ∂ w I v ( z , w ) {\displaystyle I_{v-1}(z,w)-I_{v+1}(z,w)={\dfrac {2v}{z}}I_{v}(z,w)-{\dfrac {2\tanh vw}{z}}{\dfrac {\partial }{\partial w}}I_{v}(z,w)} K v − 1 ( z , w ) − K v + 1 ( z , w ) = − 2 v z K v ( z , w ) + 2 tanh ⁡ v w z ∂ ∂ w K v ( z , w ) {\displaystyle K_{v-1}(z,w)-K_{v+1}(z,w)=-{\dfrac {2v}{z}}K_{v}(z,w)+{\dfrac {2\tanh vw}{z}}{\dfrac {\partial }{\partial w}}K_{v}(z,w)} H v − 1 ( 1 ) ( z , w ) + H v + 1 ( 1 ) ( z , w ) = 2 v z H v ( 1 ) ( z , w ) − 2 tanh ⁡ v w z ∂ ∂ w H v ( 1 ) ( z , w ) {\displaystyle H_{v-1}^{(1)}(z,w)+H_{v+1}^{(1)}(z,w)={\dfrac {2v}{z}}H_{v}^{(1)}(z,w)-{\dfrac {2\tanh vw}{z}}{\dfrac {\partial }{\partial w}}H_{v}^{(1)}(z,w)} H v − 1 ( 2 ) ( z , w ) + H v + 1 ( 2 ) ( z , w ) = 2 v z H v ( 2 ) ( z , w ) − 2 tanh ⁡ v w z ∂ ∂ w H v ( 2 ) ( z , w ) {\displaystyle H_{v-1}^{(2)}(z,w)+H_{v+1}^{(2)}(z,w)={\dfrac {2v}{z}}H_{v}^{(2)}(z,w)-{\dfrac {2\tanh vw}{z}}{\dfrac {\partial }{\partial w}}H_{v}^{(2)}(z,w)}

Where the new parameter w {\displaystyle w} defines the integral bound of the upper-incomplete form and lower-incomplete form of the modified Bessel function of the second kind:¹

K v ( z , w ) = ∫ w ∞ e − z cosh ⁡ t cosh ⁡ v t d t {\displaystyle K_{v}(z,w)=\int _{w}^{\infty }e^{-z\cosh t}\cosh vt~dt} J v ( z , w ) = ∫ 0 w e − z cosh ⁡ t cosh ⁡ v t d t {\displaystyle J_{v}(z,w)=\int _{0}^{w}e^{-z\cosh t}\cosh vt~dt}

Properties

J v ( z , w ) = J v ( z ) + e v π i 2 J ( i z , v , w ) − e − v π i 2 J ( − i z , v , w ) i π {\displaystyle J_{v}(z,w)=J_{v}(z)+{\dfrac {e^{\frac {v\pi i}{2}}J(iz,v,w)-e^{-{\frac {v\pi i}{2}}}J(-iz,v,w)}{i\pi }}} Y v ( z , w ) = Y v ( z ) + e v π i 2 J ( i z , v , w ) + e − v π i 2 J ( − i z , v , w ) π {\displaystyle Y_{v}(z,w)=Y_{v}(z)+{\dfrac {e^{\frac {v\pi i}{2}}J(iz,v,w)+e^{-{\frac {v\pi i}{2}}}J(-iz,v,w)}{\pi }}} I − v ( z , w ) = I v ( z , w ) {\displaystyle I_{-v}(z,w)=I_{v}(z,w)} for integer v {\displaystyle v} I − v ( z , w ) − I v ( z , w ) = I − v ( z ) − I v ( z ) − 2 sin ⁡ v π π J ( z , v , w ) {\displaystyle I_{-v}(z,w)-I_{v}(z,w)=I_{-v}(z)-I_{v}(z)-{\dfrac {2\sin v\pi }{\pi }}J(z,v,w)} I v ( z , w ) = I v ( z ) + J ( − z , v , w ) − e − v π i J ( z , v , w ) i π {\displaystyle I_{v}(z,w)=I_{v}(z)+{\dfrac {J(-z,v,w)-e^{-v\pi i}J(z,v,w)}{i\pi }}} I v ( z , w ) = e − v π i 2 J v ( i z , w ) {\displaystyle I_{v}(z,w)=e^{-{\frac {v\pi i}{2}}}J_{v}(iz,w)} K − v ( z , w ) = K v ( z , w ) {\displaystyle K_{-v}(z,w)=K_{v}(z,w)} K v ( z , w ) = π 2 I − v ( z , w ) − I v ( z , w ) sin ⁡ v π {\displaystyle K_{v}(z,w)={\dfrac {\pi }{2}}{\dfrac {I_{-v}(z,w)-I_{v}(z,w)}{\sin v\pi }}} for non-integer v {\displaystyle v} H v ( 1 ) ( z , w ) = J v ( z , w ) + i Y v ( z , w ) {\displaystyle H_{v}^{(1)}(z,w)=J_{v}(z,w)+iY_{v}(z,w)} H v ( 2 ) ( z , w ) = J v ( z , w ) − i Y v ( z , w ) {\displaystyle H_{v}^{(2)}(z,w)=J_{v}(z,w)-iY_{v}(z,w)} H − v ( 1 ) ( z , w ) = e v π i H v ( 1 ) ( z , w ) {\displaystyle H_{-v}^{(1)}(z,w)=e^{v\pi i}H_{v}^{(1)}(z,w)} H − v ( 2 ) ( z , w ) = e − v π i H v ( 2 ) ( z , w ) {\displaystyle H_{-v}^{(2)}(z,w)=e^{-v\pi i}H_{v}^{(2)}(z,w)} H v ( 1 ) ( z , w ) = J − v ( z , w ) − e − v π i J v ( z , w ) i sin ⁡ v π = Y − v ( z , w ) − e − v π i Y v ( z , w ) sin ⁡ v π {\displaystyle H_{v}^{(1)}(z,w)={\dfrac {J_{-v}(z,w)-e^{-v\pi i}J_{v}(z,w)}{i\sin v\pi }}={\dfrac {Y_{-v}(z,w)-e^{-v\pi i}Y_{v}(z,w)}{\sin v\pi }}} for non-integer v {\displaystyle v} H v ( 2 ) ( z , w ) = e v π i J v ( z , w ) − J − v ( z , w ) i sin ⁡ v π = Y − v ( z , w ) − e v π i Y v ( z , w ) sin ⁡ v π {\displaystyle H_{v}^{(2)}(z,w)={\dfrac {e^{v\pi i}J_{v}(z,w)-J_{-v}(z,w)}{i\sin v\pi }}={\dfrac {Y_{-v}(z,w)-e^{v\pi i}Y_{v}(z,w)}{\sin v\pi }}} for non-integer v {\displaystyle v}

Differential equations

K v ( z , w ) {\displaystyle K_{v}(z,w)} satisfies the inhomogeneous Bessel's differential equation

z 2 d 2 y d z 2 + z d y d z − ( x 2 + v 2 ) y = ( v sinh ⁡ v w + z cosh ⁡ v w sinh ⁡ w ) e − z cosh ⁡ w {\displaystyle z^{2}{\dfrac {d^{2}y}{dz^{2}}}+z{\dfrac {dy}{dz}}-(x^{2}+v^{2})y=(v\sinh vw+z\cosh vw\sinh w)e^{-z\cosh w}}

Both J v ( z , w ) {\displaystyle J_{v}(z,w)} , Y v ( z , w ) {\displaystyle Y_{v}(z,w)} , H v ( 1 ) ( z , w ) {\displaystyle H_{v}^{(1)}(z,w)} and H v ( 2 ) ( z , w ) {\displaystyle H_{v}^{(2)}(z,w)} satisfy the partial differential equation

z 2 ∂ 2 y ∂ z 2 + z ∂ y ∂ z + ( z 2 − v 2 ) y − ∂ 2 y ∂ w 2 + 2 v tanh ⁡ v w ∂ y ∂ w = 0 {\displaystyle z^{2}{\dfrac {\partial ^{2}y}{\partial z^{2}}}+z{\dfrac {\partial y}{\partial z}}+(z^{2}-v^{2})y-{\dfrac {\partial ^{2}y}{\partial w^{2}}}+2v\tanh vw{\dfrac {\partial y}{\partial w}}=0}

Both I v ( z , w ) {\displaystyle I_{v}(z,w)} and K v ( z , w ) {\displaystyle K_{v}(z,w)} satisfy the partial differential equation

z 2 ∂ 2 y ∂ z 2 + z ∂ y ∂ z − ( z 2 + v 2 ) y − ∂ 2 y ∂ w 2 + 2 v tanh ⁡ v w ∂ y ∂ w = 0 {\displaystyle z^{2}{\dfrac {\partial ^{2}y}{\partial z^{2}}}+z{\dfrac {\partial y}{\partial z}}-(z^{2}+v^{2})y-{\dfrac {\partial ^{2}y}{\partial w^{2}}}+2v\tanh vw{\dfrac {\partial y}{\partial w}}=0}

Integral representations

Base on the preliminary definitions above, one would derive directly the following integral forms of J v ( z , w ) {\displaystyle J_{v}(z,w)} , Y v ( z , w ) {\displaystyle Y_{v}(z,w)} :

J v ( z , w ) = J v ( z ) + 1 π i ( ∫ 0 w e v π i 2 − i z cosh ⁡ t cosh ⁡ v t d t − ∫ 0 w e i z cosh ⁡ t − v π i 2 cosh ⁡ v t d t ) = J v ( z ) + 1 π i ( ∫ 0 w cos ⁡ ( z cosh ⁡ t − v π 2 ) cosh ⁡ v t d t − i ∫ 0 w sin ⁡ ( z cosh ⁡ t − v π 2 ) cosh ⁡ v t d t − ∫ 0 w cos ⁡ ( z cosh ⁡ t − v π 2 ) cosh ⁡ v t d t − i ∫ 0 w sin ⁡ ( z cosh ⁡ t − v π 2 ) cosh ⁡ v t d t ) = J v ( z ) + 1 π i ( − 2 i ∫ 0 w sin ⁡ ( z cosh ⁡ t − v π 2 ) cosh ⁡ v t d t ) = J v ( z ) − 2 π ∫ 0 w sin ⁡ ( z cosh ⁡ t − v π 2 ) cosh ⁡ v t d t {\displaystyle {\begin{aligned}J_{v}(z,w)&=J_{v}(z)+{\dfrac {1}{\pi i}}\left(\int _{0}^{w}e^{{\frac {v\pi i}{2}}-iz\cosh t}\cosh vt~dt-\int _{0}^{w}e^{iz\cosh t-{\frac {v\pi i}{2}}}\cosh vt~dt\right)\\&=J_{v}(z)+{\dfrac {1}{\pi i}}\left(\int _{0}^{w}\cos \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt-i\int _{0}^{w}\sin \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt\right.\\&\quad \quad \quad \quad \quad \quad \left.-\int _{0}^{w}\cos \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt-i\int _{0}^{w}\sin \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt\right)\\&=J_{v}(z)+{\dfrac {1}{\pi i}}\left(-2i\int _{0}^{w}\sin \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt\right)\\&=J_{v}(z)-{\dfrac {2}{\pi }}\int _{0}^{w}\sin \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt\end{aligned}}} Y v ( z , w ) = Y v ( z ) + 1 π ( ∫ 0 w e v π i 2 − i z cosh ⁡ t cosh ⁡ v t d t + ∫ 0 w e i z cosh ⁡ t − v π i 2 cosh ⁡ v t d t ) = Y v ( z ) + 1 π ( ∫ 0 w cos ⁡ ( z cosh ⁡ t − v π 2 ) cosh ⁡ v t d t − i ∫ 0 w sin ⁡ ( z cosh ⁡ t − v π 2 ) cosh ⁡ v t d t + ∫ 0 w cos ⁡ ( z cosh ⁡ t − v π 2 ) cosh ⁡ v t d t + i ∫ 0 w sin ⁡ ( z cosh ⁡ t − v π 2 ) cosh ⁡ v t d t ) = Y v ( z ) + 2 π ∫ 0 w cos ⁡ ( z cosh ⁡ t − v π 2 ) cosh ⁡ v t d t {\displaystyle {\begin{aligned}Y_{v}(z,w)&=Y_{v}(z)+{\dfrac {1}{\pi }}\left(\int _{0}^{w}e^{{\frac {v\pi i}{2}}-iz\cosh t}\cosh vt~dt+\int _{0}^{w}e^{iz\cosh t-{\frac {v\pi i}{2}}}\cosh vt~dt\right)\\&=Y_{v}(z)+{\dfrac {1}{\pi }}\left(\int _{0}^{w}\cos \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt-i\int _{0}^{w}\sin \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt\right.\\&\quad \quad \quad \quad \quad \quad \left.+\int _{0}^{w}\cos \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt+i\int _{0}^{w}\sin \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt\right)\\&=Y_{v}(z)+{\dfrac {2}{\pi }}\int _{0}^{w}\cos \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt\end{aligned}}}

With the Mehler–Sonine integral expressions of J v ( z ) = 2 π ∫ 0 ∞ sin ⁡ ( z cosh ⁡ t − v π 2 ) cosh ⁡ v t d t {\displaystyle J_{v}(z)={\dfrac {2}{\pi }}\int _{0}^{\infty }\sin \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt} and Y v ( z ) = − 2 π ∫ 0 ∞ cos ⁡ ( z cosh ⁡ t − v π 2 ) cosh ⁡ v t d t {\displaystyle Y_{v}(z)=-{\dfrac {2}{\pi }}\int _{0}^{\infty }\cos \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt} mentioned in Digital Library of Mathematical Functions,²

we can further simplify to J v ( z , w ) = 2 π ∫ w ∞ sin ⁡ ( z cosh ⁡ t − v π 2 ) cosh ⁡ v t d t {\displaystyle J_{v}(z,w)={\dfrac {2}{\pi }}\int _{w}^{\infty }\sin \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt} and Y v ( z , w ) = − 2 π ∫ w ∞ cos ⁡ ( z cosh ⁡ t − v π 2 ) cosh ⁡ v t d t {\displaystyle Y_{v}(z,w)=-{\dfrac {2}{\pi }}\int _{w}^{\infty }\cos \left(z\cosh t-{\dfrac {v\pi }{2}}\right)\cosh vt~dt} , but the issue is not quite good since the convergence range will reduce greatly to | v | < 1 {\displaystyle |v|<1} .

External links

Agrest, Matest M.; Maksimov, Michail S. (1971). Theory of Incomplete Cylindrical Functions and their Applications. Berlin, Heidelberg: Springer-Verlag Berlin Heidelberg. ISBN 978-3-642-65023-9.
Cicchetti, R.; Faraone, A. (December 2004). "Incomplete Hankel and Modified Bessel Functions: A Class of Special Functions for Electromagnetics". IEEE Transactions on Antennas and Propagation. 52 (12): 3373–3389. Bibcode:2004ITAP...52.3373C. doi:10.1109/TAP.2004.835269. S2CID 25089438.
Jones, D. S. (October 2007). "Incomplete Bessel functions. II. Asymptotic expansions for large argument". Proceedings of the Edinburgh Mathematical Society. 50 (3): 711–723. doi:10.1017/S0013091505000908.

References

Jones, D. S. (February 2007). "Incomplete Bessel functions. I". Proceedings of the Edinburgh Mathematical Society. 50 (1): 173–183. doi:10.1017/S0013091505000490. https://doi.org/10.1017%2FS0013091505000490 ↩
Paris, R. B. (2010), "Bessel Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248. 978-0-521-19225-5 ↩