744 CHAPTER 36. ISOLATED SINGULARITIES AND ANALYTIC FUNCTIONS

where e(r)→ 0 as r → ∞. This used∫

∞

0 e−t2dt =

√π

2 . Now examine the first of theseintegrals. ∣∣∣∣∫

γr

eiz2dz∣∣∣∣ =

∣∣∣∣∫ π4

0ei(reit)

2rieitdt

∣∣∣∣≤ r

∫ π4

0e−r2 sin2tdt

=r2

∫ 1

0

e−r2u√

1−u2du

=r2

∫ r−(3/2)

0

1√1−u2

du+r2

(∫ 1

0

1√1−u2

)e−(r1/2)

which converges to zero as r→ ∞. Therefore, taking the limit as r→ ∞,√

π

2

(1+ i√

2

)=∫

∞

0eix2

dx

and so the Fresnel integrals are given by∫∞

0sinx2dx =

√π

2√

2=∫

∞

0cosx2dx.

The following example is one of the most interesting. By an auspicious choice of thecontour it is possible to obtain a very interesting formula for cotπz known as the MittagLeffler expansion of cotπz.

Example 36.6.7 Let γN be the contour which goes from −N− 12 −Ni horizontally to N +

12 −Ni and from there, vertically to N + 1

2 +Ni and then horizontally to −N− 12 +Ni and

finally vertically to −N− 12 −Ni. Thus the contour is a large rectangle and the direction of

integration is in the counter clockwise direction.

(−N− 12 )−Ni (N + 1

2 )−Ni

(N + 12 )+Ni(−N− 1

2 )+Ni

Consider the following integral.

IN ≡∫

γN

π cosπz(α2− z2)sinπz

dz

where α is not an integer. This will be used to verify the formula of Mittag Leffler,

1α2 +

∞

∑n=1

2α2−n2 =

π cotπα

α. (36.10)