Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA 1505 Mathematics I Revision Notes (Fourier Series and Vector Spaces) 5 Fourier Series Lecture Notes and Tutorial Question Review 1. The Formulas of Fourier Series. There are two kinds of formulas: one is for period 2 π ,theotherper iod2 L . The later one is much more general, and enough for all problems. f ( x )= a 0 + ± n =1 ( a n cos L x + b n sin L x ) , and the coefficients are given by the Euler’s formulas : a 0 = 1 2 L ² L L f ( x ) dx a n = 1 L ² L L f ( x )cos L xdx n =1 , 2 , ··· b n = 1 L ² L L f ( x )sin L n , 2 , The difficult part for fourier series is integration. 2. Trigonometric Functions. (a) The graph of cosine and sine functions. (b) Some special values of those function, for example sin 2 . (c) Some trigonometric identities. (d) Integration of these functions. (can use identities or integration by parts) 3. Parity. You need to know: (a) The advantage to consider parity Frst; (b) When f is an even function, which series will be applied? 4. Periodicity and Continuity. By deFnition, it is obvious that only periodic and piecewise continuous functions have their ±ourier series. However, for some kinds of partial functions, I mean a function is deFned on an interval, but not deFned on the other parts, then we can write out the ±ourier series of the function by using half-range expansions . Check details in Tutorial 4 Question 6. Also check tutorial notes 4 for more details about the advantage of half-range expansion compared with other expansions. 121
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
6 Vector Spaces 1. Arithmetic I skip the details here, since they are not difficult. You need to be very familiar with them. 2. Magnitude (Length) Given a vector u = u 1 i + u 2 j + u 3 k in 3-dimensional space, the length of the vector is given by ± u ± = ± u 2 1 + u 2 2 + u 2 3 . Remark : There are two kinds of notations for magnitude: ±± and || .(Th ef o rm e ri su s ed in your lecture notes; and the latter one is used in your textbook.) What’s the diFerence? No diFerence for 3-dimensional spaces. However, is more general, and it can be use in all cases to denote the length. is used only for some special spaces, for example, 3-dimensional spaces, 4-dimensional spaces, n -dimensional spaces. Application : Magnitude of a vector (like the module of a real number) does not have some good properties, for example, the function is not diFerentiable. Check Tutorial 5 Question 6. Some times we need to remove the magnitude by using the following property: ± u ± 2 = u · u .
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 9


This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online