How Can We Estimate the Integral of e^{iz^2} Over a Complex Contour?

[Brendy]

Homework Statement

I need to establish the estimate and inequality
$|\int_{C} e^{iz^{2}}dz| \leq\frac{\pi(1-e^{-R^{2}}}{4R} < \frac {\pi}{4R}$

where $C={z(t)=Re^{it},t\in[0,\fraq{\pi}{4}]$

Homework Equations

The Attempt at a Solution

I thought perhaps I could use the ML equality but the function doesn't have a global maximum. It does have a bound of 1 and -1 on the real and imaginary parts but using that as the maximum and then multiplying by the length does not give the inequality above.
I'm at a loss on how to do anything with this question. It's quite frustrating.

 

[mighty2000]

Hello! I would recommend approaching this problem by first understanding the properties of the given function, e^{iz^{2}}, and how it behaves on the given contour, C. From there, you can use the properties of integrals and inequalities to establish the desired estimate. Here are some steps you can follow:

1. Recall that the modulus (absolute value) of a complex number z = x + iy is given by |z| = sqrt(x^2 + y^2). In this case, we have z = e^{iz^{2}} = e^{i(x^2-y^2)} = cos(x^2-y^2) + i*sin(x^2-y^2). See if you can use this to determine the modulus of the function on the contour C.

2. Next, recall that the length of a contour C is given by its arc length, which in this case is R*|t|. Can you use this to determine the length of the contour C?

3. Now, you can use the properties of integrals to establish the given estimate. Recall that for a continuous function f(z) on a contour C, we have |∫f(z)dz| ≤ ∫|f(z)|dz. See if you can apply this property to the given function and contour to obtain the desired inequality.

I hope this helps you in solving the problem. Let me know if you have any further questions. Keep up the good work!