Mathematics A/3 - Ildikó Perjési-Hámori

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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

݇‫א‬Ժ

݂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


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