# standard solutions

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##### 1: 10.25 Definitions
###### §10.25(ii) StandardSolutions
The defining property of the second standard solution $K_{\nu}\left(z\right)$ of (10.25.1) is …
##### 2: 14.21 Definitions and Basic Properties
Standard solutions: the associated Legendre functions $P^{\mu}_{\nu}\left(z\right)$, $P^{-\mu}_{\nu}\left(z\right)$, $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)$, and $\boldsymbol{Q}^{\mu}_{-\nu-1}\left(z\right)$. … …
##### 3: 10.2 Definitions
###### §10.2(ii) StandardSolutions
The notation $\mathscr{C}_{\nu}\left(z\right)$ denotes $J_{\nu}\left(z\right)$, $Y_{\nu}\left(z\right)$, ${H^{(1)}_{\nu}}\left(z\right)$, ${H^{(2)}_{\nu}}\left(z\right)$, or any nontrivial linear combination of these functions, the coefficients in which are independent of $z$ and $\nu$. …
##### 4: 14.2 Differential Equations
###### §14.2(i) Legendre’s Equation
Standard solutions: $\mathsf{P}_{\nu}\left(\pm x\right)$, $\mathsf{Q}_{\nu}\left(\pm x\right)$, $\mathsf{Q}_{-\nu-1}\left(\pm x\right)$, $P_{\nu}\left(\pm x\right)$, $Q_{\nu}\left(\pm x\right)$, $Q_{-\nu-1}\left(\pm x\right)$. …
###### §14.2(ii) Associated Legendre Equation
Standard solutions: $\mathsf{P}^{\mu}_{\nu}\left(\pm x\right)$, $\mathsf{P}^{-\mu}_{\nu}\left(\pm x\right)$, $\mathsf{Q}^{\mu}_{\nu}\left(\pm x\right)$, $\mathsf{Q}^{\mu}_{-\nu-1}\left(\pm x\right)$, $P^{\mu}_{\nu}\left(\pm x\right)$, $P^{-\mu}_{\nu}\left(\pm x\right)$, $\boldsymbol{Q}^{\mu}_{\nu}\left(\pm x\right)$, $\boldsymbol{Q}^{\mu}_{-\nu-1}\left(\pm x\right)$. …
##### 6: 12.2 Differential Equations
###### §12.2 Differential Equations
Standard solutions are $U\left(a,\pm z\right)$, $V\left(a,\pm z\right)$, $\overline{U}\left(a,\pm x\right)$ (not complex conjugate), $U\left(-a,\pm iz\right)$ for (12.2.2); $W\left(a,\pm x\right)$ for (12.2.3); $D_{\nu}\left(\pm z\right)$ for (12.2.4), where …
##### 7: 13.2 Definitions and Basic Properties
###### StandardSolutions
The first two standard solutions are: … Another standard solution of (13.2.1) is $U\left(a,b,z\right)$, which is determined uniquely by the property …
13.2.7 $U\left(-m,b,z\right)=(-1)^{m}{\left(b\right)_{m}}M\left(-m,b,z\right)=(-1)^{m}% \sum_{s=0}^{m}\genfrac{(}{)}{0.0pt}{}{m}{s}{\left(b+s\right)_{m-s}}(-z)^{s}.$
13.2.8 $U\left(a,a+n+1,z\right)=\frac{(-1)^{n}{\left(1-a-n\right)_{n}}}{z^{a+n}}M\left% (-n,1-a-n,z\right)=z^{-a}\sum_{s=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{s}{\left(a% \right)_{s}}z^{-s}.$
##### 8: 14.3 Definitions and Hypergeometric Representations
14.3.8 $P^{m}_{\nu}\left(x\right)=\frac{\Gamma\left(\nu+m+1\right)}{2^{m}\Gamma\left(% \nu-m+1\right)}\left(x^{2}-1\right)^{m/2}\mathbf{F}\left(\nu+m+1,m-\nu;m+1;% \tfrac{1}{2}-\tfrac{1}{2}x\right).$
As standard solutions of (14.2.2) we take the pair $P^{-\mu}_{\nu}\left(x\right)$ and $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$, where …
##### 9: 9.2 Differential Equation
Standard solutions are: …
##### 10: 13.14 Definitions and Basic Properties
###### StandardSolutions
Standard solutions are: …