nLab Bessel function

Idea

(…)

Properties

Integral representations

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

(DLMF 10.9.9)

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

(DLMF 10.9.9)

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

(DLMF 10.32.6)

Related concepts

• special function

• causal propagator, Hadamard propagator

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

• Wikipedia, Bessel function