# scaled

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## 11—20 of 57 matching pages

##### 11: 21.4 Graphics
Figure 21.4.1 provides surfaces of the scaled Riemann theta function $\hat{\theta}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$, with … For the scaled Riemann theta functions depicted in Figures 21.4.221.4.5 Figure 21.4.4: A real-valued scaled Riemann theta function: θ ^ ⁡ ( i ⁢ x , i ⁢ y | Ω 1 ) , 0 ≤ x ≤ 4 , 0 ≤ y ≤ 4 . … Magnify 3D Help Figure 21.4.5: The real part of a genus 3 scaled Riemann theta function: ℜ ⁡ θ ^ ⁡ ( x + i ⁢ y , 0 , 0 | Ω 2 ) , 0 ≤ x ≤ 1 , 0 ≤ y ≤ 3 . … Magnify 3D Help
##### 12: 36.5 Stokes Sets
For $z\neq 0$, the Stokes set is expressed in terms of scaled coordinates …
36.5.10 $160u^{6}+40u^{4}=Y^{2}.$
36.5.17 $Y_{\mathrm{S}}(X)=Y(u,|X|),$
36.5.18 $f(u,X)=f(-u+\tfrac{1}{3},X),$
##### 13: 15.14 Integrals
15.14.1 $\int_{0}^{\infty}x^{s-1}\mathbf{F}\left({a,b\atop c};-x\right)\mathrm{d}x=% \frac{\Gamma\left(s\right)\Gamma\left(a-s\right)\Gamma\left(b-s\right)}{\Gamma% \left(a\right)\Gamma\left(b\right)\Gamma\left(c-s\right)},$ $\min(\Re a,\Re b)>\Re s>0$.
##### 14: 14.19 Toroidal (or Ring) Functions
where the constant $c$ is a scaling factor. … With $\mathbf{F}$ as in §14.3 and $\xi>0$,
14.19.2 $P^{\mu}_{\nu-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\Gamma\left(\frac{1}{2}-% \mu\right)}{\pi^{1/2}\left(1-e^{-2\xi}\right)^{\mu}e^{(\nu+(1/2))\xi}}\*% \mathbf{F}\left(\tfrac{1}{2}-\mu,\tfrac{1}{2}+\nu-\mu;1-2\mu;1-e^{-2\xi}\right),$ $\mu\neq\frac{1}{2},\frac{3}{2},\frac{5}{2},\ldots$.
14.19.3 $\boldsymbol{Q}^{\mu}_{\nu-\frac{1}{2}}\left(\cosh\xi\right)=\frac{\pi^{1/2}% \left(1-e^{-2\xi}\right)^{\mu}}{e^{(\nu+(1/2))\xi}}\*\mathbf{F}\left(\mu+% \tfrac{1}{2},\nu+\mu+\tfrac{1}{2};\nu+1;e^{-2\xi}\right).$
##### 15: 15.8 Transformations of Variable
15.8.1 $\mathbf{F}\left({a,b\atop c};z\right)=(1-z)^{-a}\mathbf{F}\left({a,c-b\atop c}% ;\frac{z}{z-1}\right)=(1-z)^{-b}\mathbf{F}\left({c-a,b\atop c};\frac{z}{z-1}% \right)=(1-z)^{c-a-b}\mathbf{F}\left({c-a,c-b\atop c};z\right),$ $|\operatorname{ph}\left(1-z\right)|<\pi$.
15.8.2 $\frac{\sin\left(\pi(b-a)\right)}{\pi}\mathbf{F}\left({a,b\atop c};z\right)=% \frac{(-z)^{-a}}{\Gamma\left(b\right)\Gamma\left(c-a\right)}\mathbf{F}\left({a% ,a-c+1\atop a-b+1};\frac{1}{z}\right)-\frac{(-z)^{-b}}{\Gamma\left(a\right)% \Gamma\left(c-b\right)}\mathbf{F}\left({b,b-c+1\atop b-a+1};\frac{1}{z}\right),$ $|\operatorname{ph}\left(-z\right)|<\pi$.
15.8.3 $\frac{\sin\left(\pi(b-a)\right)}{\pi}\mathbf{F}\left({a,b\atop c};z\right)=% \frac{(1-z)^{-a}}{\Gamma\left(b\right)\Gamma\left(c-a\right)}\mathbf{F}\left({% a,c-b\atop a-b+1};\frac{1}{1-z}\right)-\frac{(1-z)^{-b}}{\Gamma\left(a\right)% \Gamma\left(c-b\right)}\mathbf{F}\left({b,c-a\atop b-a+1};\frac{1}{1-z}\right),$ $|\operatorname{ph}\left(-z\right)|<\pi$.
Alternatively, if $b-a$ is a negative integer, then we interchange $a$ and $b$ in $\mathbf{F}\left(a,b;c;z\right)$. …
15.8.12 $\mathbf{F}\left(a,b;a+b-m;z\right)=(1-z)^{-m}\mathbf{F}\left(\tilde{a},\tilde{% b};\tilde{a}+\tilde{b}+m;z\right),$ $\tilde{a}=a-m,\tilde{b}=b-m$.
##### 16: 15.9 Relations to Other Functions
15.9.16 $\mathbf{F}\left({a,b\atop 2b};z\right)=\frac{\sqrt{\pi}}{\Gamma\left(b\right)}% z^{-b+(\ifrac{1}{2})}(1-z)^{(b-a-(\ifrac{1}{2}))/2}\*P^{-b+(\ifrac{1}{2})}_{a-% b-(\ifrac{1}{2})}\left(\frac{2-z}{2\sqrt{1-z}}\right),$ $b\neq 0,-1,-2,\dots$, $|\operatorname{ph}\left(1-z\right)|<\pi$ and $|1-z|<1$.
15.9.18 $\mathbf{F}\left({a,b\atop a+b+\tfrac{1}{2}};z\right)=2^{a+b-(\ifrac{1}{2})}(-z% )^{(-a-b+(\ifrac{1}{2}))/2}\*P^{-a-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\left% (\sqrt{1-z}\right),$ $\left|\operatorname{ph}\left(-z\right)\right|<\pi$.
15.9.19 $\mathbf{F}\left({a,b\atop a-b+1};z\right)=z^{\ifrac{(b-a)}{2}}(1-z)^{-b}\*P^{b% -a}_{-b}\left(\frac{1+z}{1-z}\right),$ $|\operatorname{ph}z|<\pi$ and $|\operatorname{ph}\left(1-z\right)|<\pi$.
15.9.20 $\mathbf{F}\left({a,b\atop\tfrac{1}{2}(a+b+1)};z\right)=\left(-z(1-z)\right)^{% \ifrac{(1-a-b)}{4}}\*P^{\ifrac{(1-a-b)}{2}}_{\ifrac{(a-b-1)}{2}}\left(1-2z% \right),$ $\left|\operatorname{ph}\left(-z\right)\right|<\pi$.
15.9.21 $\mathbf{F}\left({a,1-a\atop c};z\right)=\left(\frac{-z}{1-z}\right)^{\ifrac{(1% -c)}{2}}\*P^{1-c}_{-a}\left(1-2z\right),$ $\left|\operatorname{ph}\left(-z\right)\right|<\pi$.
##### 17: 32.13 Reductions of Partial Differential Equations
has the scaling reduction … has the scaling reduction … has the scaling reduction …
##### 18: Viewing DLMF Interactive 3D Graphics
Users can render a 3D scene and interactively rotate, scale, and otherwise explore a function surface. …
##### 19: 15.3 Graphics Figure 15.3.7: | F ⁡ ( - 3 , 3 5 ; u + i ⁢ v ; 1 2 ) | , - 6 ≤ u ≤ 2 , - 2 ≤ v ≤ 2 . Magnify 3D Help
##### 20: About Color Map
Mathematically, we scale the height to $h$ lying in the interval $[0,4]$ and the components are computed as follows … Specifically, by scaling the phase angle in $[0,2\pi)$ to $q$ in the interval $[0,4)$, the hue (in degrees) is computed as …