# Fourier sine and cosine transform formula

Jim Lambers MAT 417/517 Spring Semester 2013-14 Lecture 18 Notes These notes correspond to Lesson 25 in the text. The Finite Fourier Transforms When solving a PDE on a nite interval 0 <x<L, whether it be the heat equation or wave 4. Half Range Fourier Series. If a function is defined over half the range, say `0` to L, instead of the full range from `-L` to `L`, it may be expanded in a series of sine terms only or of cosine terms only. The series produced is then called a half range Fourier series. Fourier transform unitary, angular frequency Fourier transform unitary, ordinary frequency Remarks 10 The rectangular pulse and the normalized sinc function 11 Dual of rule 10. The rectangular function is an idealized low-pass filter, and the sinc function is the non-causal impulse response of such a filter. 12 tri is the triangular function 13 ﬁnding out what combinations of sines and cosines of varying frequencies and amplitudes will sum to the given function. This is called Fourier Analysis. 3.2 Fourier Trigonometric Series As we have seen in the last section, we are interested in ﬁnding representations of functions in terms of sines and cosines. Given a function