§8.3 Graphics

Permalink:: http://dlmf.nist.gov/8.3

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

§8.3(i) Real Variables

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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $x$ : real variable and $a$ : parameter
Permalink:: http://dlmf.nist.gov/8.3.F1
Encodings:: pdf, png

Figure 8.3.2: $\mathop{\gamma\/}\nolimits\!\left(a,x\right)$ , $a$ = 0.25, 0.5, 0.75, 1.

Symbols:: $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $x$ : real variable and $a$ : parameter
Permalink:: http://dlmf.nist.gov/8.3.F2
Encodings:: pdf, png

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

Symbols:: $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $x$ : real variable and $a$ : parameter
Permalink:: http://dlmf.nist.gov/8.3.F3
Encodings:: pdf, png

Figure 8.3.4: $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,x\right)$ (= $x^{{-a}}\mathop{P\/}\nolimits\!\left(a,x\right)$ ), $a$ = 0.25, 0.5, 0.75, 1, 2.

Symbols:: $\mathop{P\/}\nolimits\!\left(a,z\right)$ : normalized incomplete gamma function, $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $x$ : real variable and $a$ : parameter
Permalink:: http://dlmf.nist.gov/8.3.F4
Encodings:: pdf, png

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.

Symbols:: $\mathop{Q\/}\nolimits\!\left(a,z\right)$ : normalized incomplete gamma function, $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $x$ : real variable and $a$ : parameter
Permalink:: http://dlmf.nist.gov/8.3.F5
Encodings:: pdf, png

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Figure 8.3.6: $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,x\right)$ (= $x^{{-a}}\mathop{P\/}\nolimits\!\left(a,x\right)$ ), $-4\leq x\leq 4$ , $-5\leq a\leq 4$ .

Symbols:: $\mathop{P\/}\nolimits\!\left(a,z\right)$ : normalized incomplete gamma function, $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $x$ : real variable and $a$ : parameter
Referenced by:: §8.13(i), §8.3(i)
Permalink:: http://dlmf.nist.gov/8.3.F6
Encodings:: VRML, X3D, pdf, png

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).

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

Symbols:: $\mathop{Q\/}\nolimits\!\left(a,z\right)$ : normalized incomplete gamma function, $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $x$ : real variable and $a$ : parameter
Permalink:: http://dlmf.nist.gov/8.3.F7
Encodings:: VRML, X3D, pdf, png

§8.3(ii) Complex Argument

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

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.

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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 $x=y=0$ , $\mathop{\Gamma\/}\nolimits\!\left(0.25,0\right)=\mathop{\Gamma\/}\nolimits\!\left(0.25\right)=3.625\ldots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/8.3.F8
Encodings:: VRML, X3D, pdf, png

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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.

Symbols:: $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/8.3.F9
Encodings:: VRML, X3D, pdf, png

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Figure 8.3.10: $\mathop{\gamma^{{*}}\/}\nolimits\!\left(0.25,x+iy\right)$ , $-3\leq x\leq 3$ , $-3\leq y\leq 3$ .

Symbols:: $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/8.3.F10
Encodings:: VRML, X3D, pdf, png

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/8.3.F11
Encodings:: VRML, X3D, pdf, png

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Figure 8.3.12: $\mathop{\gamma\/}\nolimits\!\left(1,x+iy\right)$ , $-3\leq x\leq 3$ , $-3\leq y\leq 3$ .

Symbols:: $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/8.3.F12
Encodings:: VRML, X3D, pdf, png

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

Symbols:: $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/8.3.F13
Encodings:: VRML, X3D, pdf, png

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Figure 8.3.14: $\mathop{\Gamma\/}\nolimits\!\left(2.5,x+iy\right)$ , $-2.2\leq x\leq 3$ , $-3\leq y\leq 3$ . Principal value. There is a cut along the negative real axis. When $x=y=0$ , $\mathop{\Gamma\/}\nolimits\!\left(2.5,0\right)=\mathop{\Gamma\/}\nolimits\!\left(2.5\right)=1.329\ldots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/8.3.F14
Encodings:: VRML, X3D, pdf, png

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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.