# of two complex variables

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##### 1: 1.9 Calculus of a Complex Variable
###### Continuity
That is, given any positive number $\epsilon$, however small, we can find a positive number $\delta$ such that $\left|f(z)-f(z_{0})\right|<\epsilon$ for all $z$ in the open disk $\left|z-z_{0}\right|<\delta$. A function of two complex variables $f(z,w)$ is continuous at $(z_{0},w_{0})$ if $\lim\limits_{(z,w)\to(z_{0},w_{0})}f(z,w)=f(z_{0},w_{0})$; compare (1.5.1) and (1.5.2). …
##### 2: 16.13 Appell Functions
The following four functions of two real or complex variables $x$ and $y$ cannot be expressed as a product of two ${{}_{2}F_{1}}$ functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1): …
##### 3: 19.25 Relations to Other Functions
19.25.9 $E\left(\phi,k\right)=R_{F}\left(c-1,c-k^{2},c\right)-\tfrac{1}{3}k^{2}R_{D}% \left(c-1,c-k^{2},c\right),$
19.25.11 $E\left(\phi,k\right)=-\tfrac{1}{3}{k^{\prime}}^{2}R_{D}\left(c-k^{2},c,c-1% \right)+\ifrac{\sqrt{c-k^{2}}}{\left(\sqrt{c}\sqrt{c-1}\right)},$ $\phi\neq\tfrac{1}{2}\pi$.
19.25.13 $D\left(\phi,k\right)=\tfrac{1}{3}R_{D}\left(c-1,c-k^{2},c\right).$
19.25.21 $\operatorname{el2}\left(x,k_{c},a,b\right)=a\operatorname{el1}\left(x,k_{c}% \right)+\tfrac{1}{3}{(b-a)}R_{D}\left(r,r+k_{c}^{2},r+1\right),$
19.25.25 $(z-x)^{3/2}R_{D}\left(x,y,z\right)=(3/k^{2})(F\left(\phi,k\right)-E\left(\phi,% k\right)),$
##### 4: Mathematical Introduction
Special functions with a complex variable are depicted as colored 3D surfaces in a similar way to functions of two real variables, but with the vertical height corresponding to the modulus (absolute value) of the function. …
##### 5: 31.1 Special Notation
 $x$, $y$ real variables. complex variables. … complex parameter, $|a|\geq 1,a\neq 1$. complex parameters.
Sometimes the parameters are suppressed.
##### 6: 28.15 Expansions for Small $q$
28.15.3 $\operatorname{me}_{\nu}\left(z,q\right)=e^{\mathrm{i}\nu z}-\frac{q}{4}\left(% \frac{1}{\nu+1}e^{\mathrm{i}(\nu+2)z}-\frac{1}{\nu-1}e^{\mathrm{i}(\nu-2)z}% \right)+\frac{q^{2}}{32}\left(\frac{1}{(\nu+1)(\nu+2)}e^{\mathrm{i}(\nu+4)z}+% \frac{1}{(\nu-1)(\nu-2)}e^{\mathrm{i}(\nu-4)z}-\frac{2(\nu^{2}+1)}{(\nu^{2}-1)% ^{2}}e^{\mathrm{i}\nu z}\right)+\cdots;$
##### 7: 8.11 Asymptotic Approximations and Expansions
###### §8.11 Asymptotic Approximations and Expansions
For bounds on $R_{n}(a,z)$ when $a$ is real and $z$ is complex see Olver (1997b, pp. 109–112). … See Tricomi (1950b) for these approximations, together with higher terms and extensions to complex variables. … For (8.11.18) and extensions to complex values of $x$ see Buckholtz (1963). …
##### 8: 19.21 Connection Formulas
19.21.1 $R_{F}\left(0,z+1,z\right)R_{D}\left(0,z+1,1\right)+R_{D}\left(0,z+1,z\right)R_% {F}\left(0,z+1,1\right)=3\pi/(2z),$ $z\in\mathbb{C}\setminus(-\infty,0]$.
##### 9: 4.2 Definitions
$\ln z$ is a single-valued analytic function on $\mathbb{C}\setminus(-\infty,0]$ and real-valued when $z$ ranges over the positive real numbers. … In the DLMF we allow a further extension by regarding the cut as representing two sets of points, one set corresponding to the “upper side” and denoted by $z=x+\mathrm{i}0$, the other set corresponding to the “lower side” and denoted by $z=x-\mathrm{i}0$. …Consequently $\ln z$ is two-valued on the cut, and discontinuous across the cut. … The function $\exp$ is an entire function of $z$, with no real or complex zeros. … This is an analytic function of $z$ on $\mathbb{C}\setminus(-\infty,0]$, and is two-valued and discontinuous on the cut shown in Figure 4.2.1, unless $a\in\mathbb{Z}$. …
##### 10: Bibliography B
• E. Barouch, B. M. McCoy, and T. T. Wu (1973) Zero-field susceptibility of the two-dimensional Ising model near $T_{c}$ . Phys. Rev. Lett. 31, pp. 1409–1411.
• H. A. Bethe and E. E. Salpeter (1957) Quantum mechanics of one- and two-electron atoms. Springer-Verlag, Berlin.
• H. A. Bethe and E. E. Salpeter (1977) Quantum Mechanics of One- and Two-electron Atoms. Rosetta edition, Plenum Publishing Corp., New York.
• P. Boalch (2006) The fifty-two icosahedral solutions to Painlevé VI. J. Reine Angew. Math. 596, pp. 183–214.
• S. Bochner and W. T. Martin (1948) Several Complex Variables. Princeton Mathematical Series, Vol. 10, Princeton University Press, Princeton, N.J..