Mathematics A/3 Engineering Mathematics 3 Function series
1. Find the Fourier series generated by ݂ ሺݔሻǤ ͳ ݂݅ Ͳ ݔ൏ ߨ ݂ ሺ ݔሻ ൌ ቊ ǡ ݂ሺ ݔ ݇ʹߨሻ ൌ ݂ ሺݔሻǡ െͳ ݂݅ െ ߨ ݔ൏ Ͳ
݇אԺ
ݔ ݂݅ െ ߨ ݔ൏ ߨǡ ݂ ሺ ݔ ݇ʹߨሻ ൌ ݂ሺݔሻǡ ʹ
ߨ Ͳ ݂݅ ݔ ݔ൏ ߨ ʹ ǡ ݂ ሺ ݔሻ ൌ ൞ ߨ െ ݂݅ ݔെ ߨ ݔ൏ Ͳ ʹ
݇אԺ
݂ ሺ ݔ ݇ʹߨሻ ൌ ݂ሺݔሻǡ
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݂ሺݔሻ ൌ ȁݔȁ ݂݅ െ ߨ ݔ൏ ߨǡ ݂ ሺ ݔ ݇ʹߨሻ ൌ ݂ሺݔሻǡ ݔെ ͳ ݂݅ Ͳ ݔ൏ ߨ ǡ െͳ ݂݅ െ ߨ ݔ൏ Ͳ
݂ሺ ݔ ݇ʹߨሻ ൌ ݂ሺݔሻǡ
݇אԺ
݇אԺ
݂ ሺݔሻ ൌ ݔଶ ݂݅ െ ߨ ݔ൏ ߨǡ ݂ ሺ ݔ ݇ʹߨሻ ൌ ݂ሺݔሻǡ
݇אԺ
݂ ሺ ݔሻ ൌ
݂ ሺ ݔሻ ൌ ቊ
ߨ ߨ ۓെ ʹ ݂݅ െ ߨ ݔ൏ െ ʹ ۖ ߨ ߨ ሺ ሻ ݂݅ ݔെ ݔ൏ ǡ ݂ ݔൌ ʹ ʹ ۔ ۖ ߨ ݂݅ ߨ ݔ൏ ߨ ʹ ە ʹ ݂ ሺ ݔሻ ൌ ቊ
െʹ ݔ ͳ ݂݅ െ ߨ ݔ൏ Ͳ ǡ ʹ ݔ ͳ ݂݅ Ͳ ݔ൏ ߨ
ݔ ͳ ݂݅ െ ߨ ݔ൏ Ͳ ǡ ݂ ሺ ݔሻ ൌ ቐʹ ͳ ݂݅ Ͳ ݔ൏ ߨ
݂ ሺ ݔ ݇ʹߨሻ ൌ ݂ ሺݔሻǡ
݂ሺ ݔ ݇ʹߨሻ ൌ ݂ሺݔሻǡ ݂ ሺ ݔ ݇ʹߨሻ ൌ ݂ሺݔሻǡ
݇אԺ
݇אԺ ݇אԺ