A treatise on the theory of Bessel functions, by G. N. Watson.

INTEGRAL REPRESENTATIONS

177

the contour in (3) may be replaced by one which goes from oo - ( - ) i to
~o + (r + r)i. By taking the contour to be three sides of a rectangle with
corners at oo - (7r - ) i, - (rr - ) i, (7r + ) i, and oo + (,r + w) i, we obtain,
as a modification of (4),
(5) J (Z) e- eiZsinCosO cos (O - z cos ~r sin 0) dO
7T e o
e-vi sin r 7r - ~ —zsinh(t+i)-vt dt.
7r o
Again, if we take r to be an angle between 0 and cr, the contour in (3) may
be replaced by one which passes from cc - (~ 7w + +r)i to oo + (Ir T * #) i, and
so we find that
(6) Jv (z)- cos (v0 z sin 0) d
+ - e- z sinh tsin —vt sin (z cosh t cos - - vrr - ) dt,
7r o
provided that I arg z is less than both ~ and r- -r.
When R (v) > 0 and z is positive (=x), we may take =0 in the last
formula, and get*
(7) J, (x) = os (v - c sin 0) dO + - e-~t sin (x cosh t - v) dt.
7 J o 7rr 0
Another important formula, derived from (1), is obtained by spreading out
the contour until it is parallel to the imaginary axis on the right of the origin;
by Jordan's lemma this is permissible if R (v) > - 1, and we then obtain the
formula
(8) J ' (zi)= - 2)v t-V- exp tt- d,
in which c may have any positive value; this integral is the basis of many of
Sonine's investigations.
Integrals which resemble those given in this section are of importance in the investigation of the diffraction of light by a prism; see Carslaw, Proc. London IMlath. Soc. xxx.
(1899), pp. 121-161; W. H. Jackson, Proc. London Math. Soc. (2) I. (1904), pp. 393-414;
Whipple, Proc. London Math. Soc. (2) xvi. (1917), pp. 94-111.
6'21. Integrals which represent fnctions of the second and third kinds.
If we substitute Schliafli's integral 6'2 (4) for both of the Bessel functions
on the right of the equation
Y, (z) = J, (z) cot vTr - J_, (z) cosec v7r,
we find that,r Yv (z) = cot v7r cos (vO - z sin 0) dO - cosec v7r cos (iO + z sin 0) dO
o0
- os v7j e-vt-zsinht dt- et-zsinht dt.
* Cf. Gubler, iMath. Ann. XLIX. (1897), pp. 583-584.

W. B. F.

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