### 4.62 Double and triple integrals with constant limits

4.620 General formulas

1.

${\int }_{0}^{\pi }\text{d}\omega \text{?}{\int }_{0}^{\infty }{f}^{\prime }\text{?}\left(p\text{?}cosh\text{?}x+q\text{?}\text{cos}\text{?}\omega \text{?}\text{sinh}\text{?}x\right)\text{?}\text{sinh}\text{?}x\text{?}\text{d}x\text{?}=-\frac{\pi \text{?}\text{sign}\text{?}p}{\sqrt{{p}^{2}-{q}^{2}}}f\left(\text{sign}\text{?}p\sqrt{{p}^{2}-{q}^{2}}\right)$ LO III 389

$[p2>q2,limx→+∞f(x)=0]$ 2.

${\int }_{0}^{2\pi }\text{d}\omega {\int }_{0}^{\infty }{f}^{\prime }\left[p\text{?}cosh\text{?}x+\left(q\text{?}cos\text{?}\omega +r\text{?}sin\text{?}\omega \right)\text{?}sinh\text{?}x\right]\text{?}sinh\text{?}x\text{?}\text{d}x=-\frac{2\pi \text{?}sign\text{?}p}{\sqrt{{p}^{2}-{q}^{2}-{r}^{2}}}f\left(signp\sqrt{{p}^{2}-{q}^{2}-{r}^{2}}\right)$ LO III 390

$[??p2>q2+r2,limx→+∞f(x)=0]$ 3.

${\int }_{0}^{\pi }{\int }_{0}^{\pi }\frac{\text{d}x\text{?}\text{d}y}{}$

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