# cross-products

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## 1—10 of 17 matching pages

##### 5: 28.28 Integrals, Integral Representations, and Integral Equations
28.28.25 $\dfrac{\sinh z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\cos t\mathrm{me}_{\nu}\left(t,h% ^{2}\right)\mathrm{me}_{-\nu-2m-1}\left(t,h^{2}\right)}{{\sinh}^{2}z+{\sin}^{2% }t}\mathrm{d}t=(-1)^{m+1}\mathrm{i}h\alpha^{(0)}_{\nu,m}\mathrm{D}_{0}\left(% \nu,\nu+2m+1,z\right),$
28.28.26 $\dfrac{\cosh z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\sin t\mathrm{me}_{\nu}\left(t,h% ^{2}\right)\mathrm{me}_{-\nu-2m-1}\left(t,h^{2}\right)}{{\sinh}^{2}z+{\sin}^{2% }t}\mathrm{d}t=(-1)^{m+1}\mathrm{i}h\alpha^{(1)}_{\nu,m}\mathrm{D}_{0}\left(% \nu,\nu+2m+1,z\right),$
28.28.30 $\dfrac{\sinh z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\cos t\mathrm{me}_{\nu}'\left(t,% h^{2}\right)\mathrm{me}_{-\nu-2m-1}\left(t,h^{2}\right)}{{\sinh}^{2}z+{\sin}^{% 2}t}\mathrm{d}t=(-1)^{m}\mathrm{i}h\alpha^{(1)}_{\nu,m}\mathrm{D}_{1}\left(\nu% ,\nu+2m+1,z\right),$
###### §28.28(iv) Integrals of Products of Mathieu Functions of Integer Order
28.28.49 $\widehat{\alpha}_{n,m}^{(c)}=\frac{1}{2\pi}\int_{0}^{2\pi}\cos t\mathrm{ce}_{n% }\left(t,h^{2}\right)\mathrm{ce}_{m}\left(t,h^{2}\right)\mathrm{d}t=(-1)^{p+1}% \dfrac{2}{\mathrm{i}\pi}\dfrac{\mathrm{ce}_{n}\left(0,h^{2}\right)\mathrm{ce}_% {m}\left(0,h^{2}\right)}{h\mathrm{Dc}_{0}\left(n,m,0\right)}.$
##### 7: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
###### §28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
For further power series of Mathieu radial functions of integer order for small parameters and improved convergence rate see Larsen et al. (2009).
##### 8: 1.6 Vectors and Vector-Valued Functions
###### CrossProduct (or Vector Product)
1.6.9 $\mathbf{a}\times\mathbf{b}=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\ a_{1}&a_{2}&a_{3}\\ b_{1}&b_{2}&b_{3}\end{vmatrix}\\ =(a_{2}b_{3}-a_{3}b_{2})\mathbf{i}+(a_{3}b_{1}-a_{1}b_{3})\mathbf{j}+(a_{1}b_{% 2}-a_{2}b_{1})\mathbf{k}\\ =\|\mathbf{a}\|\|\mathbf{b}\|(\sin\theta)\mathbf{n},$
14.2.5 $\mathsf{P}^{\mu}_{\nu+1}\left(x\right)\mathsf{Q}^{\mu}_{\nu}\left(x\right)-% \mathsf{P}^{\mu}_{\nu}\left(x\right)\mathsf{Q}^{\mu}_{\nu+1}\left(x\right)=% \frac{\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+2\right)},$
14.2.11 $P^{\mu}_{\nu+1}\left(x\right)Q^{\mu}_{\nu}\left(x\right)-P^{\mu}_{\nu}\left(x% \right)Q^{\mu}_{\nu+1}\left(x\right)=e^{\mu\pi i}\frac{\Gamma\left(\nu+\mu+1% \right)}{\Gamma\left(\nu-\mu+2\right)}.$