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Bessel function

Contents

Idea

(…)

Properties

Integral representations

(1)J 0(x)=2π 0 sin(xcosh(t))dtAAAAifx>0 J_0(x) \;=\; \frac{2}{\pi} \int_0^\infty \sin\left( x \, \cosh(t) \right) \, dt \phantom{AAAA} \text{if} \,\, x \gt 0

(DLMF 10.9.9)

(2)N 0(x)Y 0(x)=2π 0 cos(xcosh(t))dtAAAAifx>0 N_0(x) \coloneqq Y_0(x) \;=\; -\frac{2}{\pi} \int_0^\infty \cos\left( x \,\cosh(t) \right) \, dt \phantom{AAAA} \text{if} \,\, x \gt 0

(DLMF 10.9.9)

(3)K 0(x)= 0 cos(xsinh(t))dtAAAAifx>0 K_0(x) \;=\; \int_0^\infty \cos\left( x \,\sinh(t) \right) \, dt \phantom{AAAA} \text{if} \,\, x \gt 0

(DLMF 10.32.6)

References

  • Milton Abramowitz, Irene Stegun, sections 9, 10, 11 of_Handbook of mathematical functions_, 1964 (pdf)

    chapter 9: F. W. J. Olver, Bessel functions of integer order (pdf)

  • G. N. Watson, A treatise on the theory of Bessel functions, Cambridge University Press 1966 (web)

  • Digital Library of Mathematical Functions, chapter 10 Bessel functions

See also

Last revised on November 26, 2017 at 18:04:06. See the history of this page for a list of all contributions to it.