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6 Exponential, Logarithmic, Sine, and Cosine IntegralsProperties

§6.7 Integral Representations

Contents

§6.7(i) Exponential Integrals

6.7.1 0e-att+bdt=0eiatt+ibdt=eabE1(ab),
a>0, b>0,
6.7.2 ex0αe-xt1-tdt=Ei(x)-Ei((1-α)x),
0α<1, x>0.
6.7.3 xeita2+t2dt=i2a(eaE1(a-ix)-e-aE1(-a-ix)),
a>0, x>0,
6.7.4 xteita2+t2dt=12(eaE1(a-ix)+e-aE1(-a-ix)),
a>0, x>0.
6.7.5 xe-ta2+t2dt=-12ai(eiaE1(x+ia)-e-iaE1(x-ia)),
a>0, x,
6.7.6 xte-ta2+t2dt=12(eiaE1(x+ia)+e-iaE1(x-ia)),
a>0, x.
6.7.7 01e-atsin(bt)tdt=Ein(a+ib),
a,b,
6.7.8 01e-at(1-cos(bt))tdt=Ein(a+ib)-Ein(a),
a,b.

Many integrals with exponentials and rational functions, for example, integrals of the type ezR(z)dz, where R(z) is an arbitrary rational function, can be represented in finite form in terms of the function E1(z) and elementary functions; see Lebedev (1965, p. 42).

§6.7(ii) Sine and Cosine Integrals

When z

6.7.9 si(z)=-0π/2e-zcostcos(zsint)dt,
6.7.10 Ein(z)-Cin(z)=0π/2e-zcostsin(zsint)dt,
6.7.11 01(1-e-at)cos(bt)tdt=Ein(a+ib)-Cin(b),
a,b.

§6.7(iii) Auxiliary Functions

6.7.12 g(z)+if(z)=e-izzeittdt,
|phz|π.

The path of integration does not cross the negative real axis or pass through the origin.

6.7.13 f(z) =0sintt+zdt
=0e-ztt2+1dt,
6.7.14 g(z) =0costt+zdt
=0te-ztt2+1dt.

The first integrals on the right-hand sides apply when |phz|<π; the second ones when z0 and (in the case of (6.7.14)) z0.

§6.7(iv) Compendia

For collections of integral representations see Bierens de Haan (1939, pp. 56–59, 72–73, 82–84, 121, 133–136, 155, 179–181, 223, 225–227, 230, 259–260, 374, 377, 397–398, 408, 416, 424, 431, 438–439, 442–444, 488, 496–500, 567–571, 585, 602, 638, 675–677), Corrington (1961), Erdélyi et al. (1954a, vol. 1, pp. 267–270), Geller and Ng (1969), Nielsen (1906b), Oberhettinger (1974, pp. 244–246), Oberhettinger and Badii (1973, pp. 364–371), and Watrasiewicz (1967).