Complex Analysis: Theory and Applications · Problem 7.2.16

Taylor and Laurent Series — Howell & Matthews

Mathematics — Complex Analysis · taylor-series-fresnel-integrals

Problem

The Fresnel integrals

C(z)

and

S(z)

are defined by

C(z) = \int_0^z \cos(\xi^2)\,d\xi \quad\text{and}\quad S(z) = \int_0^z \sin(\xi^2)\,d\xi.

Define

F(z)

F(z) = C(z) + iS(z)

and complete the following: ((a)) Verify the identity

F(z) = \int_0^z \exp(i\xi^2)\,d\xi

. ((b)) Integrate the power series for

\exp(i\xi^2)

and obtain the power series for

F(z)

. ((c)) Use the partial sum involving terms up to

z^9

to find approximations to

C(1.0)

and

S(1.0)