# §28.10 Integral Equations

## §28.10(i) Equations with Elementary Kernels

With the notation of §28.4 for Fourier coefficients,

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

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

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

## §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).