EMLecture4

# EMLecture4 - Lecture 4 Notes 95.657 Electromagnetic Theory...

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

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Lecture 4 Notes, 95.657 Electromagnetic Theory I, Fall 2011 Dr. Christopher S. Baird, UMass Lowell 1. Orthogonal Functions and Expansions- In the interval ( a , b ) of the variable x , a set of real or complex functions U n ( x ) where n = 1, 2, ... are orthogonal if: ∫ a b U n * x U m x dx = 0, m ≠ n- When m = n , the integral is nonzero. The functions are orthonormal if normalized to one: ∫ a b U n * x U m x dx = nm- An arbitrary, integrable function f ( x ) can be expanded in a series of the orthonormal functions U n ( x ) according to: f x = ∑ n = 1 N a n U n x - To find the expansion coefficients a n , we multiply both sides by the function U * M ( x ), integrate, and use the orthonormality property: f x U m * x = ∑ n = 1 N a n U n x U m * x ∫ a b f x U m * x dx = ∑ n = 1 N a n ∫ a b U n x U m * x dx ∫ a b f x U m * x dx = ∑ n = 1 N a n nm ∫ a b f x U m * x dx = a m- Interchange the arbitrary label m for n and get the final form: f ( x )= ∑ n = 1 N a n U n ( x ) where a n = ∫ a b f ( x ) U n * ( x ) dx Finite Series Expansion- If the functions form a complete set, and all functions that are useful in physics do, then the series expansion becomes a more accurate representation of the function f ( x ) as more terms in the series are kept. The most accurate is the infinite series: f ( x )= ∑ n = 1 ∞ a n U n ( x ) where a n = ∫ a b f ( x ) U n * ( x ) dx Infinite Series Expansion- If the interval ( a , b ) is expanded to be infinite, than the orthogonal functions become a continuum of functions, the index variable n becomes a continuous variable k , and the orthogonality condition becomes normalized to the Dirac delta function: ∫ −∞ ∞ U k * x U k ' x dx = k − k ' f ( x )= ∫ −∞ ∞ A ( k ) U k ( x ) dk where A ( k )= ∫ −∞ ∞ f ( x ) U n * ( x ) dx Infinite Continuous Expansion 2. Fourier Series- The most commonly used orthogonal functions are sines and cosines, constituting Fourier series.- Start with a general expansion in terms of sines and cosines over the interval (- a /2, a /2): f x = ∑ n = ∞ A n cos k n x B n sin k n x - In order for the series to be a valid representation of the function in the interval, the series must be periodic outside the interval, so that f (- a /2) = f ( a /2). Using this requirement leads to: ∑ n = ∞ A n cos k n − a / 2 − cos k n a / 2 B n sin k n − a / 2 − sin k n a / 2 =- This must be true independent of A n and B n , so that the coefficients must be zero: sin k n − a / 2 − sin k n a / 2 = sin k n a / 2 = k n a / 2 = n k n = 2 n a The series now becomes: f x = ∑ n = ∞ A n cos 2 n x a B n sin 2 n x a To find the coefficients, multiply both sides by...
View Full Document

{[ snackBarMessage ]}

### Page1 / 14

EMLecture4 - Lecture 4 Notes 95.657 Electromagnetic Theory...

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

View Full Document
Ask a homework question - tutors are online