11.7. EXERCISES 273
complex, explain why sin(z) cannot be bounded. Hint: Use the theorem about thezero set having a limit point on a connected open set.
14. The functions z→ sin(z2),z→ cos
(z2), and z→ eiz2
are all analytic functions sincethe chain rule continues to hold for functions of a complex variable. This problem ison the Fresnel integrals using contour integrals. In this case, there is no singular partof the function. The contour to use is
x
y
Then using this contour and the integral∫
∞
0 e−t2dt =
√π
2 , explain why
0 =∫
γr
eiz2dz+
∫ r
0eix2
dx−∫ r
0ei(
t(
1+i√2
))2(1+ i√2
)dt
=∫
γr
eiz2dz+
∫ r
0eix2
dx−∫ r
0e−t2
(1+ i√
2
)dt
=∫
γr
eiz2dz+
∫ r
0eix2
dx−√
π
2
(1+ i√
2
)+ e(r)
Where limr→0 e(r) = 0. Now examine the first integral. Explain the following stepsand why this integral converges to 0 as r→ ∞.∣∣∣∣∫
γr
eiz2dz∣∣∣∣= ∣∣∣∣∫ π
4
0ei(reit)
2rieitdt
∣∣∣∣≤ r∫ π
4
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)
15. If γ : C→ C is a parametrization of a curve with γ being differentiable, one to oneon (a,b) ,a < b with continuous derivative, the length of C is defined as
sup
{∑P|γ (ti)− γ (ti−1)| ,P a partition of [a,b]
}Show this is independent of equivalent smooth parametrization and that in every case,it equals
∫ ba |γ ′ (t)|dt, the integral of the absolute value of the derivative.
16. Consider the following contour consisting of the orientation shown by the arrows.
γr
γR
There is a large semicircle on the top of radius R and a small one of radius r. If γ refersto the piecewise smooth, oriented contour consisting of the two straight lines and two