# several variables

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##### 11: 1.14 Integral Transforms
The same notation $\mathscr{F}$ is used for Fourier transforms of functions of several variables and for Fourier transforms of distributions; see §1.16(vii). …
##### 12: Bibliography B
• S. Bochner and W. T. Martin (1948) Several Complex Variables. Princeton Mathematical Series, Vol. 10, Princeton University Press, Princeton, N.J..
• ##### 13: 2.3 Integrals of a Real Variable
###### §2.3 Integrals of a Real Variable
$k$ ($\leq\infty$) and $\lambda$ are positive constants, $\alpha$ is a variable parameter in an interval $\alpha_{1}\leq\alpha\leq\alpha_{2}$ with $\alpha_{1}\leq 0$ and $0<\alpha_{2}\leq k$, and $x$ is a large positive parameter. …In consequence, the approximation is nonuniform with respect to $\alpha$ and deteriorates severely as $\alpha\to 0$. A uniform approximation can be constructed by quadratic change of integration variable:
##### 14: Annie A. M. Cuyt
Subsequently she was a Research fellow with the Alexander von Humboldt Foundation (Germany), she obtained the Habilitation (1986) and became author or co-author of several books, including Handbook of Continued Fractions for Special Functions. …A lot of her research has been devoted to rational approximations, in one as well as in many variables, and sparse interpolation. …
##### 15: 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 $\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{a-1}F\left(a,b;c;z% \right)\right)={\left(a\right)_{n}}z^{a+n-1}F\left(a+n,b;c;z\right).$
Other versions of several of the identities in this subsection can be constructed with the aid of the operator identity
15.5.10 $\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}=z^{n}\frac{{\mathrm{d}}^{n}}% {{\mathrm{d}z}^{n}}z^{n},$ $n=1,2,3,\dots$.
##### 16: 13.3 Recurrence Relations and Derivatives
13.3.7 $U\left(a-1,b,z\right)+(b-2a-z)U\left(a,b,z\right)+a(a-b+1)U\left(a+1,b,z\right% )=0,$
13.3.8 $(b-a-1)U\left(a,b-1,z\right)+(1-b-z)U\left(a,b,z\right)+zU\left(a,b+1,z\right)% =0,$
13.3.9 $U\left(a,b,z\right)-aU\left(a+1,b,z\right)-U\left(a,b-1,z\right)=0,$
13.3.10 $(b-a)U\left(a,b,z\right)+U\left(a-1,b,z\right)-zU\left(a,b+1,z\right)=0,$
Other versions of several of the identities in this subsection can be constructed with the aid of the operator identity …
##### 17: 19.16 Definitions
A fourth integral that is symmetric in only two variables is defined by
19.16.5 $R_{D}\left(x,y,z\right)=R_{J}\left(x,y,z,z\right)=\frac{3}{2}\int_{0}^{\infty}% \frac{\mathrm{d}t}{s(t)(t+z)},$
Thus $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$ is symmetric in the variables $z_{j}$ and $z_{\ell}$ if the parameters $b_{j}$ and $b_{\ell}$ are equal. The $R$-function is often used to make a unified statement of a property of several elliptic integrals. … When one variable is 0 without destroying convergence, any one of (19.16.14)–(19.16.17) is said to be complete and can be written as an $R$-function with one less variable: …
##### 18: 5.4 Special Values and Extrema
5.4.16 $\Im\psi\left(iy\right)=\frac{1}{2y}+\frac{\pi}{2}\coth\left(\pi y\right),$
5.4.17 $\Im\psi\left(\tfrac{1}{2}+iy\right)=\frac{\pi}{2}\tanh\left(\pi y\right),$
##### 19: Errata
• Subsection 1.16(vii)

Several changes have been made to

1. (i)

make consistent use of the Fourier transform notations $\mathscr{F}\left(f\right)$, $\mathscr{F}\left(\phi\right)$ and $\mathscr{F}\left(u\right)$ where $f$ is a function of one real variable, $\phi$ is a test function of $n$ variables associated with tempered distributions, and $u$ is a tempered distribution (see (1.14.1), (1.16.29) and (1.16.35));

2. (ii)

introduce the partial differential operator $\mathbf{D}$ in (1.16.30);

3. (iii)

clarify the definition (1.16.32) of the partial differential operator $P(\mathbf{D})$; and

4. (iv)

clarify the use of $P(\mathbf{D})$ and $P(\mathbf{x})$ in (1.16.33), (1.16.34), (1.16.36) and (1.16.37).

• ##### 20: Frank W. J. Olver
, the behavior of solutions as the independent variable, or some parameter, tends to infinity, and in the study of the particular solutions of differential equations known as special functions (e. … He also served on the Editorial Boards of several of the leading journals devoted to mathematical and/or numerical analysis, including founding Managing Editor of the SIAM Journal on Mathematical Analysis. …