# §28.10 Integral Equations

## §28.10(i) Equations with Elementary Kernels

With the notation of §28.4 for Fourier coefficients,

 28.10.1 $\displaystyle\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\cos\left(2h\cos z\cos t% \right)\operatorname{ce}_{2n}\left(t,h^{2}\right)\,\mathrm{d}t$ $\displaystyle=\frac{A_{0}^{2n}(h^{2})}{\operatorname{ce}_{2n}\left(\frac{1}{2}% \pi,h^{2}\right)}\operatorname{ce}_{2n}\left(z,h^{2}\right),$ 28.10.2 $\displaystyle\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\cosh\left(2h\sin z\sin t% \right)\operatorname{ce}_{2n}\left(t,h^{2}\right)\,\mathrm{d}t$ $\displaystyle=\frac{A_{0}^{2n}(h^{2})}{\operatorname{ce}_{2n}\left(0,h^{2}% \right)}\operatorname{ce}_{2n}\left(z,h^{2}\right),$ 28.10.3 $\displaystyle\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\sin\left(2h\cos z\cos t% \right)\operatorname{ce}_{2n+1}\left(t,h^{2}\right)\,\mathrm{d}t$ $\displaystyle=-\frac{hA_{1}^{2n+1}(h^{2})}{\operatorname{ce}_{2n+1}'\left(% \frac{1}{2}\pi,h^{2}\right)}\operatorname{ce}_{2n+1}\left(z,h^{2}\right),$
 28.10.4 $\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\cos z\cos t\cosh\left(2h\sin z\sin t% \right)\operatorname{ce}_{2n+1}\left(t,h^{2}\right)\,\mathrm{d}t=\frac{A_{1}^{% 2n+1}(h^{2})}{2\operatorname{ce}_{2n+1}\left(0,h^{2}\right)}\operatorname{ce}_% {2n+1}\left(z,h^{2}\right),$
 28.10.5 $\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\sinh\left(2h\sin z\sin t\right)% \operatorname{se}_{2n+1}\left(t,h^{2}\right)\,\mathrm{d}t=\frac{hB_{1}^{2n+1}(% h^{2})}{\operatorname{se}_{2n+1}'\left(0,h^{2}\right)}\operatorname{se}_{2n+1}% \left(z,h^{2}\right),$
 28.10.6 $\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\sin z\sin t\cos\left(2h\cos z\cos t% \right)\operatorname{se}_{2n+1}\left(t,h^{2}\right)\,\mathrm{d}t=\frac{B_{1}^{% 2n+1}(h^{2})}{2\operatorname{se}_{2n+1}\left(\frac{1}{2}\pi,h^{2}\right)}% \operatorname{se}_{2n+1}\left(z,h^{2}\right),$
 28.10.7 $\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\sin z\sin t\sin\left(2h\cos z\cos t% \right)\operatorname{se}_{2n+2}\left(t,h^{2}\right)\,\mathrm{d}t=-\frac{hB_{2}% ^{2n+2}(h^{2})}{2\operatorname{se}_{2n+2}'\left(\frac{1}{2}\pi,h^{2}\right)}% \operatorname{se}_{2n+2}\left(z,h^{2}\right),$
 28.10.8 $\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\cos z\cos t\sinh\left(2h\sin z\sin t% \right)\operatorname{se}_{2n+2}\left(t,h^{2}\right)\,\mathrm{d}t=\frac{hB_{2}^% {2n+2}(h^{2})}{2\operatorname{se}_{2n+2}'\left(0,h^{2}\right)}\operatorname{se% }_{2n+2}\left(z,h^{2}\right).$

## §28.10(ii) Equations with Bessel-Function Kernels

 28.10.9 $\displaystyle\int_{0}^{\ifrac{\pi}{2}}J_{0}\left(2\sqrt{q({\cos}^{2}\tau-{\sin% }^{2}\zeta)}\right)\operatorname{ce}_{2n}\left(\tau,q\right)\,\mathrm{d}\tau$ $\displaystyle=w_{\mbox{\tiny II}}(\tfrac{1}{2}\pi;a_{2n}\left(q\right),q)% \operatorname{ce}_{2n}\left(\zeta,q\right),$ 28.10.10 $\displaystyle\int_{0}^{\pi}J_{0}\left(2\sqrt{q}(\cos\tau+\cos\zeta)\right)% \operatorname{ce}_{n}\left(\tau,q\right)\,\mathrm{d}\tau$ $\displaystyle=w_{\mbox{\tiny II}}(\pi;a_{n}\left(q\right),q)\operatorname{ce}_% {n}\left(\zeta,q\right).$

## §28.10(iii) Further Equations

See §28.28. See also Prudnikov et al. (1990, pp. 359–368), Erdélyi et al. (1955, p. 115), and Gradshteyn and Ryzhik (2000, pp. 755–759). For relations with variable boundaries see Volkmer (1983).