# complex

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##### 1: 31.3 Basic Solutions
31.3.5 $z^{1-\gamma}\mathit{H\!\ell}\left(a,(a\delta+\epsilon)(1-\gamma)+q;\alpha+1-% \gamma,\beta+1-\gamma,2-\gamma,\delta;z\right).$
31.3.6 $\mathit{H\!\ell}\left(1-a,\alpha\beta-q;\alpha,\beta,\delta,\gamma;1-z\right),$
31.3.7 $(1-z)^{1-\delta}\*\mathit{H\!\ell}\left(1-a,((1-a)\gamma+\epsilon)(1-\delta)+% \alpha\beta-q;\alpha+1-\delta,\beta+1-\delta,2-\delta,\gamma;1-z\right).$
31.3.8 $\mathit{H\!\ell}\left(\frac{a}{a-1},\frac{\alpha\beta a-q}{a-1};\alpha,\beta,% \epsilon,\delta;\frac{a-z}{a-1}\right),$
##### 2: 20.12 Mathematical Applications
###### §20.12(ii) Uniformization and Embedding of Complex Tori
For the terminology and notation see McKean and Moll (1999, pp. 48–53). The space of complex tori $\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})$ (that is, the set of complex numbers $z$ in which two of these numbers $z_{1}$ and $z_{2}$ are regarded as equivalent if there exist integers $m,n$ such that $z_{1}-z_{2}=m+\tau n$) is mapped into the projective space $P^{3}$ via the identification $z\to(\theta_{1}\left(2z\middle|\tau\right),\theta_{2}\left(2z\middle|\tau% \right),\theta_{3}\left(2z\middle|\tau\right),\theta_{4}\left(2z\middle|\tau% \right))$. Thus theta functions “uniformize” the complex torus. This ability to uniformize multiply-connected spaces (manifolds), or multi-sheeted functions of a complex variable (Riemann (1899), Rauch and Lebowitz (1973), Siegel (1988)) has led to applications in string theory (Green et al. (1988a, b), Krichever and Novikov (1989)), and also in statistical mechanics (Baxter (1982)). …
##### 3: 31.1 Special Notation
 $x$, $y$ real variables. complex variables. … complex parameter, $|a|\geq 1,a\neq 1$. complex parameters.
##### 4: 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 $\frac{\mathbf{F}\left(a,b;c;z\right)}{\mathbf{F}\left(a+1,b+1;c+1;z\right)}={x% _{0}+\cfrac{y_{1}}{x_{1}+\cfrac{y_{2}}{x_{2}+\cfrac{y_{3}}{x_{3}+\cdots}}}},$
##### 5: 4.34 Derivatives and Differential Equations
4.34.7 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-a^{2}w=0,$
4.34.8 $\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}-a^{2}w^{2}=1,$
##### 6: 15.5 Derivatives and Contiguous Functions
15.5.1 $\frac{\mathrm{d}}{\mathrm{d}z}F\left(a,b;c;z\right)=\frac{ab}{c}F\left(a+1,b+1% ;c+1;z\right),$
15.5.2 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}F\left(a,b;c;z\right)=\frac{{\left(a% \right)_{n}}{\left(b\right)_{n}}}{{\left(c\right)_{n}}}\*F\left(a+n,b+n;c+n;z% \right).$
15.5.3 $\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{a-1}F\left(a,b;c;z% \right)\right)={\left(a\right)_{n}}z^{a+n-1}F\left(a+n,b;c;z\right).$
15.5.4 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{c-1}F\left(a,b;c;z\right)% \right)={\left(c-n\right)_{n}}z^{c-n-1}F\left(a,b;c-n;z\right).$
15.5.5 $\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{c-a-1}(1-z)^{a+b-c}F% \left(a,b;c;z\right)\right)={\left(c-a\right)_{n}}z^{c-a+n-1}(1-z)^{a-n+b-c}\*% F\left(a-n,b;c;z\right).$
##### 8: 4.20 Derivatives and Differential Equations
4.20.9 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+a^{2}w=0,$
4.20.10 $\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}+a^{2}w^{2}=1,$
4.20.11 $\frac{\mathrm{d}w}{\mathrm{d}z}-a^{2}w^{2}=1,$
##### 9: 31.12 Confluent Forms of Heun’s Equation
31.12.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\frac{\gamma}{z}+\frac{% \delta}{z-1}+\epsilon\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\alpha z-q}{% z(z-1)}w=0.$
31.12.2 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\frac{\delta}{z^{2}}+\frac{% \gamma}{z}+1\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\alpha z-q}{z^{2}}w=0.$
31.12.3 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-\left(\frac{\gamma}{z}+\delta+z% \right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\alpha z-q}{z}w=0.$
31.12.4 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\gamma+z\right)z\frac{% \mathrm{d}w}{\mathrm{d}z}+\left(\alpha z-q\right)w=0.$
##### 10: 30.6 Functions of Complex Argument
###### §30.6 Functions of Complex Argument
The solutions …of (30.2.1) with $\mu=m$ and $\lambda=\lambda^{m}_{n}\left(\gamma^{2}\right)$ are real when $z\in(1,\infty)$, and their principal values (§4.2(i)) are obtained by analytic continuation to $\mathbb{C}\setminus(-\infty,1]$. … For results for Equation (30.2.1) with complex parameters see Meixner and Schäfke (1954).