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8 Incomplete Gamma and Related FunctionsIncomplete Gamma Functions

§8.3 Graphics

Contents

§8.3(i) Real Variables

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Figure 8.3.1: Γ(a,x), a = 0.25, 1, 2, 2.5, 3. Magnify
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Figure 8.3.2: γ(a,x), a = 0.25, 0.5, 0.75, 1. Magnify
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Figure 8.3.3: γ(a,x), a = 1, 2, 2.5, 3. Magnify
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Figure 8.3.4: γ*(a,x) (= x-aP(a,x)), a = 0.25, 0.5, 0.75, 1, 2. Magnify
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Figure 8.3.5: x-a-γ*(a,x) (= x-aQ(a,x)), a = 0.25, 0.5, 1, 2. Magnify 3D Help
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Figure 8.3.6: γ*(a,x) (= x-aP(a,x)), -4x4, -5a4. 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).

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Figure 8.3.7: x-a-γ*(a,x) (= x-aQ(a,x)), 0x4, -5a5. Magnify 3D Help

§8.3(ii) Complex Argument

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: Γ(0.25,x+iy), -3x3, -3y3. 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
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Figure 8.3.9: γ(0.25,x+iy), -3x3, -3y3. Principal value. There is a cut along the negative real axis. Magnify 3D Help
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Figure 8.3.10: γ*(0.25,x+iy), -3x3, -3y3. Magnify 3D Help
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Figure 8.3.11: Γ(1,x+iy), -3x3, -3y3. Magnify 3D Help
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Figure 8.3.12: γ(1,x+iy), -3x3, -3y3. Magnify 3D Help
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Figure 8.3.13: γ*(1,x+iy), -3x3, -3y3. Magnify 3D Help
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Figure 8.3.14: Γ(2.5,x+iy), -2.2x3, -3y3. 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
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Figure 8.3.15: γ(2.5,x+iy), -2.2x3, -3y3. Principal value. There is a cut along the negative real axis. Magnify 3D Help
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Figure 8.3.16: γ*(2.5,x+iy), -3x3, -3y3. Magnify 3D Help