Wronskians

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1: 10.28 Wronskians and Cross-Products
§10.28 Wronskians and Cross-Products
10.28.1 $\mathscr{W}\left\{I_{\nu}\left(z\right),I_{-\nu}\left(z\right)\right\}=I_{\nu}% \left(z\right)I_{-\nu-1}\left(z\right)-I_{\nu+1}\left(z\right)I_{-\nu}\left(z% \right)=-2\sin\left(\nu\pi\right)/(\pi z),$
10.28.2 $\mathscr{W}\left\{K_{\nu}\left(z\right),I_{\nu}\left(z\right)\right\}=I_{\nu}% \left(z\right)K_{\nu+1}\left(z\right)+I_{\nu+1}\left(z\right)K_{\nu}\left(z% \right)=1/z.$
2: 10.5 Wronskians and Cross-Products
§10.5 Wronskians and Cross-Products
10.5.1 $\mathscr{W}\left\{J_{\nu}\left(z\right),J_{-\nu}\left(z\right)\right\}=J_{\nu+% 1}\left(z\right)J_{-\nu}\left(z\right)+J_{\nu}\left(z\right)J_{-\nu-1}\left(z% \right)=-2\sin\left(\nu\pi\right)/(\pi z),$
10.5.2 $\mathscr{W}\left\{J_{\nu}\left(z\right),Y_{\nu}\left(z\right)\right\}=J_{\nu+1% }\left(z\right)Y_{\nu}\left(z\right)-J_{\nu}\left(z\right)Y_{\nu+1}\left(z% \right)=2/(\pi z),$
10.5.3 $\mathscr{W}\left\{J_{\nu}\left(z\right),{H^{(1)}_{\nu}}\left(z\right)\right\}=% J_{\nu+1}\left(z\right){H^{(1)}_{\nu}}\left(z\right)-J_{\nu}\left(z\right){H^{% (1)}_{\nu+1}}\left(z\right)=2i/(\pi z),$
10.5.4 $\mathscr{W}\left\{J_{\nu}\left(z\right),{H^{(2)}_{\nu}}\left(z\right)\right\}=% J_{\nu+1}\left(z\right){H^{(2)}_{\nu}}\left(z\right)-J_{\nu}\left(z\right){H^{% (2)}_{\nu+1}}\left(z\right)=-2i/(\pi z),$
3: 10.50 Wronskians and Cross-Products
§10.50 Wronskians and Cross-Products
$\mathscr{W}\left\{\mathsf{j}_{n}\left(z\right),\mathsf{y}_{n}\left(z\right)% \right\}=z^{-2},$
$\mathscr{W}\left\{{\mathsf{h}^{(1)}_{n}}\left(z\right),{\mathsf{h}^{(2)}_{n}}% \left(z\right)\right\}=-2iz^{-2}.$
$\mathscr{W}\left\{{\mathsf{i}^{(1)}_{n}}\left(z\right),{\mathsf{i}^{(2)}_{n}}% \left(z\right)\right\}=(-1)^{n+1}z^{-2},$
$\mathscr{W}\left\{{\mathsf{i}^{(1)}_{n}}\left(z\right),\mathsf{k}_{n}\left(z% \right)\right\}=\mathscr{W}\left\{{\mathsf{i}^{(2)}_{n}}\left(z\right),\mathsf% {k}_{n}\left(z\right)\right\}\\ =-\tfrac{1}{2}\pi z^{-2}.$
4: 14.2 Differential Equations
§14.2(iv) Wronskians and Cross-Products
14.2.3 $\mathscr{W}\left\{\mathsf{P}^{-\mu}_{\nu}\left(x\right),\mathsf{P}^{-\mu}_{\nu% }\left(-x\right)\right\}=\frac{2}{\Gamma\left(\mu-\nu\right)\Gamma\left(\nu+% \mu+1\right)\left(1-x^{2}\right)},$
14.2.6 $\mathscr{W}\left\{\mathsf{P}^{-\mu}_{\nu}\left(x\right),\mathsf{Q}^{\mu}_{\nu}% \left(x\right)\right\}=\frac{\cos\left(\mu\pi\right)}{1-x^{2}},$
14.2.7 $\mathscr{W}\left\{P^{-\mu}_{\nu}\left(x\right),P^{\mu}_{\nu}\left(x\right)% \right\}=\mathscr{W}\left\{\mathsf{P}^{-\mu}_{\nu}\left(x\right),\mathsf{P}^{% \mu}_{\nu}\left(x\right)\right\}=\frac{2\sin\left(\mu\pi\right)}{\pi\left(1-x^% {2}\right)},$
14.2.10 $\mathscr{W}\left\{P^{\mu}_{\nu}\left(x\right),Q^{\mu}_{\nu}\left(x\right)% \right\}=-e^{\mu\pi i}\frac{\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+% 1\right)\left(x^{2}-1\right)},$
5: 9.11 Products
§9.11(ii) Wronskian
9.11.2 $\mathscr{W}\left\{{\operatorname{Ai}}^{2}\left(z\right),\operatorname{Ai}\left% (z\right)\operatorname{Bi}\left(z\right),{\operatorname{Bi}}^{2}\left(z\right)% \right\}=2\pi^{-3}.$
9.11.9 $\int zw_{1}w^{\prime}_{2}\,\mathrm{d}z=\tfrac{1}{2}w^{\prime}_{1}w^{\prime}_{2% }+\tfrac{1}{4}z^{2}\mathscr{W}\left\{w_{1},w_{2}\right\},$
9.11.11 $\int\frac{1}{w_{1}^{2}}f^{\prime}\!\left(\frac{w_{2}}{w_{1}}\right)\,\mathrm{d% }z=\frac{1}{\mathscr{W}\left\{w_{1},w_{2}\right\}}f\left(\frac{w_{2}}{w_{1}}% \right).$
6: 1.13 Differential Equations
Wronskian
The Wronskian of $w_{1}(z)$ and $w_{2}(z)$ is defined by …(More generally $\mathscr{W}\left\{w_{1}(z),\ldots,w_{n}(z)\right\}=\det\left[w_{k}^{(j-1)}(z)\right]$, where $1\leq j,k\leq n$.) …If $f(z)=0$, then the Wronskian is constant. The following three statements are equivalent: $w_{1}(z)$ and $w_{2}(z)$ comprise a fundamental pair in $D$; $\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}$ does not vanish in $D$; $w_{1}(z)$ and $w_{2}(z)$ are linearly independent, that is, the only constants $A$ and $B$ such that …
8: 30.5 Functions of the Second Kind
30.5.4 $\mathscr{W}\left\{\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right),\mathsf{Qs}^{m}% _{n}\left(x,\gamma^{2}\right)\right\}=\frac{(n+m)!}{(1-x^{2})(n-m)!}A_{n}^{m}(% \gamma^{2})A_{n}^{-m}(\gamma^{2})\quad(\neq 0),$
9: 12.2 Differential Equations
§12.2(iii) Wronskians
12.2.10 $\mathscr{W}\left\{U\left(a,z\right),V\left(a,z\right)\right\}=\sqrt{2/\pi},$
12.2.11 $\mathscr{W}\left\{U\left(a,z\right),U\left(a,-z\right)\right\}=\frac{\sqrt{2% \pi}}{\Gamma\left(\frac{1}{2}+a\right)},$
12.2.12 $\mathscr{W}\left\{U\left(a,z\right),U\left(-a,\pm iz\right)\right\}=\mp ie^{% \pm i\pi(\frac{1}{2}a+\frac{1}{4})}.$
10: 13.14 Definitions and Basic Properties
§13.14(vi) Wronskians
13.14.25 $\mathscr{W}\left\{M_{\kappa,\mu}\left(z\right),M_{\kappa,-\mu}\left(z\right)% \right\}=-2\mu,$
13.14.26 $\mathscr{W}\left\{M_{\kappa,\mu}\left(z\right),W_{\kappa,\mu}\left(z\right)% \right\}=-\frac{\Gamma\left(1+2\mu\right)}{\Gamma\left(\frac{1}{2}+\mu-\kappa% \right)},$
13.14.28 $\mathscr{W}\left\{M_{\kappa,-\mu}\left(z\right),W_{\kappa,\mu}\left(z\right)% \right\}=-\frac{\Gamma\left(1-2\mu\right)}{\Gamma\left(\frac{1}{2}-\mu-\kappa% \right)},$
13.14.30 $\mathscr{W}\left\{W_{\kappa,\mu}\left(z\right),W_{-\kappa,\mu}\left(e^{\pm\pi% \mathrm{i}}z\right)\right\}=e^{\mp\kappa\pi\mathrm{i}}.$