Bessel function
Bessel's integrals

Another definition of the Bessel function, for integer values of n, is possible using an integral representation: J n (x) = 1 π ∫ 0 π cos (n τ − x sin τ) d τ = 1 π Re (∫ 0 π e i (n τ − x sin τ) d τ), {\displaystyle J_{n}(x)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(n\tau -x\sin \tau)\,d\tau ={\frac {1}{\pi }}\operatorname {Re} \left(\int _{0}^{\pi }e^{i(n\tau -x\sin \tau)}\,d\tau \right),} which is also called Hansen-Bessel formula.

This was the approach that Bessel used, and from this definition he derived several properties of the function. The definition may be extended to non-integer orders by one of Schläfli's integrals, for Re(x) > 0: J α (x) = 1 π ∫ 0 π cos (α τ − x sin τ) d τ − sin (α π) π ∫ 0 ∞ e − x sinh t − α t d t. {\displaystyle J_{\alpha }(x)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(\alpha \tau -x\sin \tau)\,d\tau -{\frac {\sin(\alpha \pi)}{\pi }}\int _{0}^{\infty }e^{-x\sinh t-\alpha t}\,dt.}
combination:two_responses