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8 Incomplete Gamma and Related FunctionsRelated Functions

§8.19 Generalized Exponential Integral


§8.19(i) Definition and Integral Representations

For p,z

8.19.1 Ep(z)=zp-1Γ(1-p,z).

Most properties of Ep(z) follow straightforwardly from those of Γ(a,z). For an extensive treatment of E1(z) see Chapter 6.

8.19.2 Ep(z)=zp-1ze-ttpdt.

When the path of integration excludes the origin and does not cross the negative real axis (8.19.2) defines the principal value of Ep(z), and unless indicated otherwise in the DLMF principal values are assumed.

Other Integral Representations

Integral representations of Mellin–Barnes type for Ep(z) follow immediately from (8.6.11), (8.6.12), and (8.19.1).

§8.19(ii) Graphics

Figure 8.19.1: Ep(x), 0x3, 0p8. Magnify

In Figures, height corresponds to the absolute value of the function and color to the phase. See About Color Map.

Figure 8.19.2: E12(x+iy), -4x4, -4y4. Principal value. There is a branch cut along the negative real axis. Magnify
Figure 8.19.3: E1(x+iy), -4x4, -4y4. Principal value. There is a branch cut along the negative real axis. Magnify
Figure 8.19.4: E32(x+iy), -3x3, -3y3. Principal value. There is a branch cut along the negative real axis. Magnify
Figure 8.19.5: E2(x+iy), -3x3, -3y3. Principal value. There is a branch cut along the negative real axis. Magnify

§8.19(iii) Special Values

8.19.5 E0(z)=z-1e-z,
8.19.6 Ep(0)=1p-1,
8.19.7 En(z)=(-z)n-1(n-1)!E1(z)+e-z(n-1)!k=0n-2(n-k-2)!(-z)k,

§8.19(iv) Series Expansions

For n=1,2,3,,

8.19.8 En(z)=(-z)n-1(n-1)!(ψ(n)-lnz)-k=0kn-1(-z)kk!(1-n+k),


8.19.9 En(z)=(-1)nzn-1(n-1)!lnz+e-z(n-1)!k=1n-1(-z)k-1Γ(n-k)+e-z(-z)n-1(n-1)!k=0zkk!ψ(k+1),

with |phz|π in both equations. For ψ(x) see §5.2(i).

When p

8.19.10 Ep(z)=zp-1Γ(1-p)-k=0(-z)kk!(1-p+k),
8.19.11 Ep(z)=Γ(1-p)(zp-1-e-zk=0zkΓ(2-p+k)),

again with |phz|π in both equations. The right-hand sides are replaced by their limiting forms when p=1,2,3,.

§8.19(v) Recurrence Relation and Derivatives

8.19.12 pEp+1(z)+zEp(z)=e-z.
8.19.13 ddzEp(z) =-Ep-1(z),
8.19.14 ddz(ezEp(z)) =ezEp(z)(1+p-1z)-1z.


For j=1,2,3,,

8.19.15 jEp(z)pj=(-1)j1(lnt)jt-pe-ztdt,

For properties and numerical tables see Milgram (1985), and also (when p=1) MacLeod (2002b).

§8.19(vi) Relation to Confluent Hypergeometric Function

8.19.16 Ep(z)=zp-1e-zU(p,p,z).

For U(a,b,z) see §13.2(i).

§8.19(vii) Continued Fraction

8.19.17 Ep(z)=e-z(1z+p1+1z+p+11+2z+),

See also Cuyt et al. (2008, pp. 277–285).

§8.19(viii) Analytic Continuation

The general function Ep(z) is attained by extending the path in (8.19.2) across the negative real axis. Unless p is a nonpositive integer, Ep(z) has a branch point at z=0. For z0 each branch of Ep(z) is an entire function of p.

8.19.18 Ep(ze2mπi)=2πiempπiΓ(p)sin(mpπ)sin(pπ)zp-1+Ep(z),
m, z0.

§8.19(ix) Inequalities

For n=1,2,3, and x>0,

8.19.19 n-1nEn(x)<En+1(x)<En(x),
8.19.20 (En(x))2<En-1(x)En+1(x),
8.19.21 1x+n<exEn(x)1x+n-1,
8.19.22 ddxEn(x)En-1(x)>0.

§8.19(x) Integrals

8.19.23 zEp-1(t)dt=Ep(z),
8.19.24 0e-atEn(t)dt=(-1)n-1an(ln(1+a)+k=1n-1(-1)kakk),
n=1,2,, a>-1,
8.19.25 0e-attb-1Ep(t)dt=Γ(b)(1+a)-bp+b-1F(1,b;p+b;a/(1+a)),
a>-1, (p+b)>1.
8.19.26 0Ep(t)Eq(t)dt=L(p)+L(q)p+q-1,
p>0, q>0, p+q>1,


8.19.27 L(p)=0e-tEp(t)dt=12pF(1,1;1+p;12),

For the hypergeometric function F(a,b;c;z) see §15.2(i). When p=1,2,3,, L(p) can also be evaluated via (8.19.24).

For collections of integrals involving Ep(z), especially for integer p, see Apelblat (1983, §§7.1–7.2) and LeCaine (1945).

§8.19(xi) Further Generalizations

For higher-order generalized exponential integrals see Meijer and Baken (1987) and Milgram (1985).