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8 Incomplete Gamma and Related Functions Incomplete Gamma Functions 8.2 Definitions and Basic Properties 8.4 Special Values

§8.3 Graphics

Permalink:: http://dlmf.nist.gov/8.3
See also:: Annotations for 8

Contents

§8.3(i) Real Variables
§8.3(ii) Complex Argument

§8.3(i) Real Variables

Keywords:: incomplete gamma functions
Notes:: These graphics were produced at NIST.
Permalink:: http://dlmf.nist.gov/8.3.i
See also:: Annotations for 8.3

See accompanying text — Figure 8.3.1: $\mathop{\Gamma\/}\nolimits\!\left(a,x\right)$ , $a$ = 0.25, 1, 2, 2.5, 3.

Magnify — Figure 8.3.1: $\mathop{\Gamma\/}\nolimits\!\left(a,x\right)$ , $a$ = 0.25, 1, 2, 2.5, 3.

See accompanying text — Figure 8.3.3: $\mathop{\gamma\/}\nolimits\!\left(a,x\right)$ , $a$ = 1, 2, 2.5, 3.

Magnify — Figure 8.3.3: $\mathop{\gamma\/}\nolimits\!\left(a,x\right)$ , $a$ = 1, 2, 2.5, 3.

See accompanying text — Figure 8.3.5: $x^{-a}-\mathop{\gamma^{*}\/}\nolimits\!\left(a,x\right)$ (= $x^{-a}\mathop{Q\/}\nolimits\!\left(a,x\right)$ ), $a$ = 0.25, 0.5, 1, 2.

Magnify — Figure 8.3.5: $x^{-a}-\mathop{\gamma^{*}\/}\nolimits\!\left(a,x\right)$ (= $x^{-a}\mathop{Q\/}\nolimits\!\left(a,x\right)$ ), $a$ = 0.25, 0.5, 1, 2.

Some monotonicity properties of $\mathop{\gamma^{*}\/}\nolimits\!\left(a,x\right)$ and $\mathop{\Gamma\/}\nolimits\!\left(a,x\right)$ in the four quadrants of the ( $a, x$ )-plane in Figure 8.3.6 are given in Erdélyi et al. (1953b, §9.6).

See accompanying text — Figure 8.3.7: $x^{-a}-\mathop{\gamma^{*}\/}\nolimits\!\left(a,x\right)$ (= $x^{-a}\mathop{Q\/}\nolimits\!\left(a,x\right)$ ), $0\leq x\leq 4$ , $-5\leq a\leq 5$ .

Magnify — Figure 8.3.7: $x^{-a}-\mathop{\gamma^{*}\/}\nolimits\!\left(a,x\right)$ (= $x^{-a}\mathop{Q\/}\nolimits\!\left(a,x\right)$ ), $0\leq x\leq 4$ , $-5\leq a\leq 5$ .

§8.3(ii) Complex Argument

Keywords:: incomplete gamma functions
Notes:: These graphics were produced at NIST.
Permalink:: http://dlmf.nist.gov/8.3.ii
See also:: Annotations for 8.3

In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the phase. See About Color Map.

See accompanying text — Figure 8.3.8: $\mathop{\Gamma\/}\nolimits\!\left(0.25,x+iy\right)$ , $-3\leq x\leq 3$ , $-3\leq y\leq 3$ . Principal value. There is a cut along the negative real axis. When $x=y=0$ , $\mathop{\Gamma\/}\nolimits\!\left(0.25,0\right)=\mathop{\Gamma\/}\nolimits\!% \left(0.25\right)=3.625\ldots$ .

Magnify — Figure 8.3.8: $\mathop{\Gamma\/}\nolimits\!\left(0.25,x+iy\right)$ , $-3\leq x\leq 3$ , $-3\leq y\leq 3$ . Principal value. There is a cut along the negative real axis. When $x=y=0$ , $\mathop{\Gamma\/}\nolimits\!\left(0.25,0\right)=\mathop{\Gamma\/}\nolimits\!% \left(0.25\right)=3.625\ldots$ .

See accompanying text — Figure 8.3.9: $\mathop{\gamma\/}\nolimits\!\left(0.25,x+iy\right)$ , $-3\leq x\leq 3$ , $-3\leq y\leq 3$ . Principal value. There is a cut along the negative real axis.

Magnify — Figure 8.3.9: $\mathop{\gamma\/}\nolimits\!\left(0.25,x+iy\right)$ , $-3\leq x\leq 3$ , $-3\leq y\leq 3$ . Principal value. There is a cut along the negative real axis.

See accompanying text — Figure 8.3.11: $\mathop{\Gamma\/}\nolimits\!\left(1,x+iy\right)$ , $-3\leq x\leq 3$ , $-3\leq y\leq 3$ .

Magnify — Figure 8.3.11: $\mathop{\Gamma\/}\nolimits\!\left(1,x+iy\right)$ , $-3\leq x\leq 3$ , $-3\leq y\leq 3$ .

See accompanying text — Figure 8.3.13: $\mathop{\gamma^{*}\/}\nolimits\!\left(1,x+iy\right)$ , $-3\leq x\leq 3$ , $-3\leq y\leq 3$ .

Magnify — Figure 8.3.13: $\mathop{\gamma^{*}\/}\nolimits\!\left(1,x+iy\right)$ , $-3\leq x\leq 3$ , $-3\leq y\leq 3$ .

See accompanying text — Figure 8.3.15: $\mathop{\gamma\/}\nolimits\!\left(2.5,x+iy\right)$ , $-2.2\leq x\leq 3$ , $-3\leq y\leq 3$ . Principal value. There is a cut along the negative real axis.

Magnify — Figure 8.3.15: $\mathop{\gamma\/}\nolimits\!\left(2.5,x+iy\right)$ , $-2.2\leq x\leq 3$ , $-3\leq y\leq 3$ . Principal value. There is a cut along the negative real axis.

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