Sonine formula

The Sonine formula refers to one of several integral representations of Bessel functions of the first kind that were derived by the Russian mathematician Nikolay Alladiyevich Sonin (1849–1915). The most widely cited form expresses the Bessel function $J_{ u}(z)$ as an integral over a finite interval involving a cosine kernel:

$$ J_{ u}(z)=\frac{\left(\dfrac{z}{2}\right)^{ u}}{\sqrt{\pi},\Gamma!\left( u+\tfrac12\right)} \int_{-1}^{1}!(1-t^{2})^{ u-\tfrac12}\cos (zt),dt ,\qquad \Re( u)>-\tfrac12 . $$

Equivalently, using the substitution $t=\cos\theta$ one obtains

$$ J_{ u}(z)=\frac{(z/2)^{ u}}{\Gamma( u+1)}\int_{0}^{\pi}!\sin^{2 u}!\theta;\cos (z\cos\theta),d\theta , $$

which is sometimes called Sonine’s integral representation.

Historical background

Nikolay Sonin investigated special functions and their integral transforms in the late 19th and early 20th centuries. His work on Bessel functions yielded the above representation, which provides a bridge between Bessel functions and elementary trigonometric integrals. The formula has been employed in the derivation of series expansions, asymptotic approximations, and in the proof of orthogonality relations for Bessel functions.

Mathematical properties

• Domain of validity – The integral converges for $\Re( u)>-1/2$; analytic continuation extends the result to other values of $ u$ where the Bessel function itself is defined.
• Symmetry – Because the integrand is an even function of $t$, the integral can be written as twice the integral from 0 to 1.
• Connection with other representations – The Sonine formula is related to the Schlömilch and Poisson integral representations of Bessel functions and can be derived from the generating function $\exp!\bigl(\tfrac{z}{2}(t-1/t)\bigr)$ by contour deformation.

Applications

• Solution of differential equations – Integral representations such as Sonine’s are useful in solving boundary‑value problems where separation of variables leads to Bessel functions.
• Transform methods – The formula underlies certain integral transforms (e.g., the Sonine–Weber transform) that map functions on $[0,\infty)$ to Bessel‑function spaces.
• Asymptotic analysis – By applying the method of stationary phase to the Sonine integral, one obtains the classical asymptotic expansion of $J_{ u}(z)$ for large $z$.

Related concepts

• Sonine kernel – The weight $(1-t^{2})^{ u-\frac12}$ appearing in the integrand is sometimes referred to as a Sonine kernel and plays a role in convolution-type identities involving Bessel functions.
• Sonine–Polubarinov formula – An extension of Sonine’s representation that incorporates products of Bessel functions.

References

1. N. A. Sonin, Ueber die Bessel'schen Funktionen, Journal für die reine und angewandte Mathematik, vol. 87 (1878), pp. 122–142.
2. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, 1944.
3. A. Erdélyi (ed.), Higher Transcendental Functions, vol. II, McGraw‑Hill, 1953 – Chapter on Bessel functions, §7.12.

The Sonine formula remains a standard tool in mathematical analysis, particularly in contexts where integral representations facilitate analytic continuation, numerical evaluation, or the derivation of further identities involving Bessel functions.