# of a complex variable

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##### 1: 12.4 Power-Series Expansions
12.4.3 $u_{1}(a,z)=e^{-\tfrac{1}{4}z^{2}}\left(1+(a+\tfrac{1}{2})\frac{z^{2}}{2!}+(a+% \tfrac{1}{2})(a+\tfrac{5}{2})\frac{z^{4}}{4!}+\cdots\right),$
12.4.5 $u_{1}(a,z)=e^{\tfrac{1}{4}z^{2}}\left(1+(a-\tfrac{1}{2})\frac{z^{2}}{2!}+(a-% \tfrac{1}{2})(a-\tfrac{5}{2})\frac{z^{4}}{4!}+\cdots\right),$
12.4.6 $u_{2}(a,z)=e^{\tfrac{1}{4}z^{2}}\left(z+(a-\tfrac{3}{2})\frac{z^{3}}{3!}+(a-% \tfrac{3}{2})(a-\tfrac{7}{2})\frac{z^{5}}{5!}+\cdots\right).$
##### 2: 15.7 Continued Fractions
15.7.1 $\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a,b+1;c+1;z\right)}=t_{0% }-\cfrac{u_{1}z}{t_{1}-\cfrac{u_{2}z}{t_{2}-\cfrac{u_{3}z}{t_{3}-\cdots}}},$
15.7.3 $\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a,b+1;c+1;z\right)}=v_{0% }-\cfrac{w_{1}}{v_{1}-\cfrac{w_{2}}{v_{2}-\cfrac{w_{3}}{v_{3}-\cdots}}},$
15.7.5 $\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a+1,b+1;c+1;z\right)}={x% _{0}+\cfrac{y_{1}}{x_{1}+\cfrac{y_{2}}{x_{2}+\cfrac{y_{3}}{x_{3}+\cdots}}}},$
##### 3: 15.5 Derivatives and Contiguous Functions
15.5.11 $(c-a)F\left(a-1,b;c;z\right)+\left(2a-c+(b-a)z\right)F\left(a,b;c;z\right)+a(z% -1)F\left(a+1,b;c;z\right)=0,$
15.5.12 $(b-a)F\left(a,b;c;z\right)+aF\left(a+1,b;c;z\right)-bF\left(a,b+1;c;z\right)=0,$
15.5.13 $(c-a-b)F\left(a,b;c;z\right)+a(1-z)F\left(a+1,b;c;z\right)-(c-b)F\left(a,b-1;c% ;z\right)=0,$
15.5.14 $c\left(a+(b-c)z\right)F\left(a,b;c;z\right)-ac(1-z)F\left(a+1,b;c;z\right)+(c-% a)(c-b)zF\left(a,b;c+1;z\right)=0,$
15.5.15 $(c-a-1)F\left(a,b;c;z\right)+aF\left(a+1,b;c;z\right)-(c-1)F\left(a,b;c-1;z% \right)=0,$
##### 4: 5.1 Special Notation
 $j,m,n$ nonnegative integers. … real or complex variables with $|q|<1$. …
##### 5: 8.8 Recurrence Relations and Derivatives
8.8.3 $w(a+2,z)-(a+1+z)w(a+1,z)+azw(a,z)=0.$
8.8.11 $P\left(a+n,z\right)=P\left(a,z\right)-z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{z^{k}}{% \Gamma\left(a+k+1\right)},$
8.8.12 $Q\left(a+n,z\right)=Q\left(a,z\right)+z^{a}e^{-z}\sum_{k=0}^{n-1}\frac{z^{k}}{% \Gamma\left(a+k+1\right)}.$
8.8.15 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}(z^{-a}\gamma\left(a,z\right))=(-1)^% {n}z^{-a-n}\gamma\left(a+n,z\right),$
##### 6: 15.1 Special Notation
15.1.1 ${{}_{2}F_{1}}\left(a,b;c;z\right)=F\left(a,b;c;z\right)=F\left({a,b\atop c};z% \right),$
##### 7: 12.8 Recurrence Relations and Derivatives
12.8.1 $zU\left(a,z\right)-U\left(a-1,z\right)+(a+\tfrac{1}{2})U\left(a+1,z\right)=0,$
12.8.2 $U'\left(a,z\right)+\tfrac{1}{2}zU\left(a,z\right)+(a+\tfrac{1}{2})U\left(a+1,z% \right)=0,$
12.8.3 $U'\left(a,z\right)-\tfrac{1}{2}zU\left(a,z\right)+U\left(a-1,z\right)=0,$
12.8.4 $2U'\left(a,z\right)+U\left(a-1,z\right)+(a+\tfrac{1}{2})U\left(a+1,z\right)=0.$
12.8.5 $zV\left(a,z\right)-V\left(a+1,z\right)+(a-\tfrac{1}{2})V\left(a-1,z\right)=0,$
##### 8: 31.17 Physical Applications
31.17.2 $\frac{x_{s}^{2}}{z_{k}}+\frac{x_{t}^{2}}{z_{k}-1}+\frac{x_{u}^{2}}{z_{k}-a}=0,$ $k=1,2$,
31.17.4 $\Psi(\mathbf{x})=(z_{1}z_{2})^{-s-\frac{1}{4}}((z_{1}-1)(z_{2}-1))^{-t-\frac{1% }{4}}\*((z_{1}-a)(z_{2}-a))^{-u-\frac{1}{4}}w(z_{1})w(z_{2}),$
##### 9: 15.11 Riemann’s Differential Equation
15.11.3 $w=P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.$
15.11.4 $w=P\begin{Bmatrix}0&1&\infty&\\ 0&0&a&z\\ 1-c&c-a-b&b&\end{Bmatrix}$
15.11.6 $P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}=P\begin{Bmatrix}\widetilde{\alpha}&\widetilde{% \beta}&\widetilde{\gamma}&\\ a_{1}&b_{1}&c_{1}&t\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.$
15.11.7 $P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}=\left(\frac{z-\alpha}{z-\gamma}\right)^{a_{1}}% \left(\frac{z-\beta}{z-\gamma}\right)^{b_{1}}P\begin{Bmatrix}0&1&\infty&\\ 0&0&a_{1}+b_{1}+c_{1}&\dfrac{(z-\alpha)(\beta-\gamma)}{(z-\gamma)(\beta-\alpha% )}\\ a_{2}-a_{1}&b_{2}-b_{1}&a_{1}+b_{1}+c_{2}&\end{Bmatrix}.$
15.11.8 $z^{\lambda}(1-z)^{\mu}P\begin{Bmatrix}0&1&\infty&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}=P\begin{Bmatrix}0&1&\infty&\\ a_{1}+\lambda&b_{1}+\mu&c_{1}-\lambda-\mu&z\\ a_{2}+\lambda&b_{2}+\mu&c_{2}-\lambda-\mu&\end{Bmatrix},$
##### 10: 15.10 Hypergeometric Differential Equation
15.10.1 $z(1-z)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(c-(a+b+1)z\right)\frac% {\mathrm{d}w}{\mathrm{d}z}-abw=0.$
15.10.3 $\mathscr{W}\left\{f_{1}(z),f_{2}(z)\right\}=(1-c)z^{-c}(1-z)^{c-a-b-1}.$
15.10.5 $\mathscr{W}\left\{f_{1}(z),f_{2}(z)\right\}=(a+b-c)z^{-c}(1-z)^{c-a-b-1}.$
15.10.7 $\mathscr{W}\left\{f_{1}(z),f_{2}(z)\right\}=(a-b)z^{-c}(z-1)^{c-a-b-1}.$
15.10.21 $w_{1}(z)=\frac{\Gamma\left(c\right)\Gamma\left(c-a-b\right)}{\Gamma\left(c-a% \right)\Gamma\left(c-b\right)}w_{3}(z)+\frac{\Gamma\left(c\right)\Gamma\left(a% +b-c\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}w_{4}(z),$