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DLMF: §8.3 Graphics ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions
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8 Incomplete Gamma and Related FunctionsIncomplete Gamma Functions

§8.3 Graphics

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Permalink:
http://dlmf.nist.gov/8.3
See also:
Annotations for Ch.8
Contents
  1. §8.3(i) Real Variables
  2. §8.3(ii) Complex Argument

§8.3(i) Real Variables

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Keywords:
graphics, incomplete gamma functions, monotonicity properties, real variables
Notes:
These graphics were produced at NIST.
Permalink:
http://dlmf.nist.gov/8.3.i
See also:
Annotations for §8.3 and Ch.8
See accompanying text
Figure 8.3.1: Γ⁡(a,x), a = 0.25, 1, 2, 2.5, 3. Magnify
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Symbols:
Γ⁡(a,z): incomplete gamma function, x: real variable and a: parameter
Permalink:
http://dlmf.nist.gov/8.3.F1
Encodings:
pdf, png
See also:
Annotations for §8.3(i), §8.3 and Ch.8
See accompanying text
Figure 8.3.2: γ⁡(a,x), a = 0.25, 0.5, 0.75, 1. Magnify
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Symbols:
γ⁡(a,z): incomplete gamma function, x: real variable and a: parameter
Permalink:
http://dlmf.nist.gov/8.3.F2
Encodings:
pdf, png
See also:
Annotations for §8.3(i), §8.3 and Ch.8
See accompanying text
Figure 8.3.3: γ⁡(a,x), a = 1, 2, 2.5, 3. Magnify
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Symbols:
γ⁡(a,z): incomplete gamma function, x: real variable and a: parameter
Permalink:
http://dlmf.nist.gov/8.3.F3
Encodings:
pdf, png
See also:
Annotations for §8.3(i), §8.3 and Ch.8
See accompanying text
Figure 8.3.4: γ∗⁡(a,x) (= x−a⁢P⁡(a,x)), a = 0.25, 0.5, 0.75, 1, 2. Magnify
See accompanying text
Figure 8.3.5: x−a−γ∗⁡(a,x) (= x−a⁢Q⁡(a,x)), a = 0.25, 0.5, 1, 2. Magnify 3D Help
See accompanying text
Figure 8.3.6: γ∗⁡(a,x) (= x−a⁢P⁡(a,x)), −4≤x≤4, −5≤a≤4. Magnify 3D Help

Some monotonicity properties of γ∗⁡(a,x) and Γ⁡(a,x) 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−γ∗⁡(a,x) (= x−a⁢Q⁡(a,x)), 0≤x≤4, −5≤a≤5. Magnify 3D Help

§8.3(ii) Complex Argument

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Keywords:
complex argument, graphics, incomplete gamma functions
Notes:
These graphics were produced at NIST.
Permalink:
http://dlmf.nist.gov/8.3.ii
See also:
Annotations for §8.3 and Ch.8

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: Γ⁡(0.25,x+i⁢y), −3≤x≤3, −3≤y≤3. Principal value. There is a cut along the negative real axis. When x=y=0, Γ⁡(0.25,0)=Γ⁡(0.25)=3.625⁢…. Magnify 3D Help
See accompanying text
Figure 8.3.9: γ⁡(0.25,x+i⁢y), −3≤x≤3, −3≤y≤3. Principal value. There is a cut along the negative real axis. Magnify 3D Help
See accompanying text
Figure 8.3.10: γ∗⁡(0.25,x+i⁢y), −3≤x≤3, −3≤y≤3. Magnify 3D Help
See accompanying text
Figure 8.3.11: Γ⁡(1,x+i⁢y), −3≤x≤3, −3≤y≤3. Magnify 3D Help
See accompanying text
Figure 8.3.12: γ⁡(1,x+i⁢y), −3≤x≤3, −3≤y≤3. Magnify 3D Help
See accompanying text
Figure 8.3.13: γ∗⁡(1,x+i⁢y), −3≤x≤3, −3≤y≤3. Magnify 3D Help
See accompanying text
Figure 8.3.14: Γ⁡(2.5,x+i⁢y), −2.2≤x≤3, −3≤y≤3. Principal value. There is a cut along the negative real axis. When x=y=0, Γ⁡(2.5,0)=Γ⁡(2.5)=1.329⁢…. Magnify 3D Help
See accompanying text
Figure 8.3.15: γ⁡(2.5,x+i⁢y), −2.2≤x≤3, −3≤y≤3. Principal value. There is a cut along the negative real axis. Magnify 3D Help
See accompanying text
Figure 8.3.16: γ∗⁡(2.5,x+i⁢y), −3≤x≤3, −3≤y≤3. Magnify 3D Help
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