Solved: Sketch or graph f(x), of period 27T, which for -7T

Chapter 11, Problem 11.1.7

(choose chapter or problem)

Euler Formulas in Terms of Jumps Without Integration. Show that for a function whose third derivative is identically zero,

\(a_{n}=\frac{1}{n \pi}\left[-\sum j_{s} \sin n x_{s}\right.-\frac{1}{n} \sum j_{s}^{\prime} \cos n x_{s}\)

\(\left.+\frac{1}{n^{2}} \sum j_{s}^{\prime \prime} \sin n x_{s}\right]\)

\(b_{n}=\frac{1}{n \pi}\left[\sum j_{s} \cos n x_{s}\right.-\frac{1}{n} \sum j_{s}^{\prime} \sin n x_{s}\)

\(\left.-\frac{1}{n^{2}} \sum j_{s}^{\prime \prime} \cos n x_{s}\right]\)

where \(n=1,2, \cdots\) and we sum over all the jumps \(j_{s}\), \(j_{s}^{\prime}, j_{s}^{\prime \prime}\) of f, f ‘, f ‘. respectively. located at \(x_{s}\).

Text Transcription:

a_n = 1 / n pi [-sum j_s sin nx_{s}. -1 / n sum j_s’ cos nx_s

+ 1 / n^2 sum j_s” sin nx_s]

b_n = 1 / n pi [sum j_s cos nx_s. -1 / n sum j_s’ sin nx_s

- 1 / n^2 sum j_s”  cos nx_s]

n = 1, 2, cdots

j_s

j_s’ , j_s”

x_s