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

8-61]

FUNCTIONS OF LARGE ORDER

265

The domains of values of the complex 7 - a + i/, for which
1 - 3 cot / - a tanh a
is positive (in the strip 0 < /3 < vr) are numbered 1, 4, 5 in Fig. 21; in the
domains numbered 2, 3, 6a, 6b, 7a, 7b the expression is negative; the corresponding domains for the complex v/z - cosh (a + i/3) have the same numbers
in Fig. 22.
4 2
TT2
Ti / \
5 4
3 2
0 / 7b 7a
Fig. 21. Fig. 22.
(I) When 1 - / cot / - a tanh a is positive, () ('u, - ) is essentially positive,
so that the curve never crosses the line v = - /. The only possibility therefore
is that the curve after crossing the real axis goes off to - oo as shewn by the
upper dotted curve in Fig. 20.
(II) When 1 -/3cot/3- a tanh a is negative, the equation qb(-a,v) =0
has no real root between 0 and 3 - 27r, for
bf (- a, v)/av = cosh a (cos / - cos v).
Therefore (-a, v) has a single maximum at -/3, and its value there is
negative, so that a (-a, v) is negative when v lies between 0 and 3 - 27r.
Also 4q (, /3 - 27r) has a maximum at iu = a, and its value there is negative,
so that the curve 0 (,u, v) = 0 does not cross v = 3 - 27r; hence, after crossing
the real axis, the curve must pass off to oo - ri, as shewn by the dotted curve
on the right of Fig. 20.
This completes the discussion of the part of the curve associated with
SV (1 (z) when a >0, 0< / t7r.
Next we have to consider what happens to the curve after crossing the
line v = + 7r.
Since 4 (a, v) = cosh a {(v -/3) cos /3 - sin v + sin /3,
and the expression on the right is positive when v > /3, the curve never crosses
the line u = a; also

) (u, n7r) = (u - a) sinh a sin / + (nTr - /) cosh a cos /3 + cosh a sin /3,