91Ó°ÊÓ

A statement, rule, or law that specifies the relationship between the independent and dependent variables is known as a function in mathematics. There are functions everywhere in mathematics, and they are essential for developing connections between the disciplines.

Given the function of the problem 9:

$f\left(x\right)=\left\{\begin{array}{c}x\\ -2\end{array}\begin{array}{ccc}0& <x& <1\\ 1& <x& <2\end{array}\right\}$

The graph of the function with at least three periods that represents the cosine series is shown in the below graph:

As the graph of the function is having even extension (observed by the above graph).

Thus, the exponential series will converge to it.

Then the graph of the of the exponential Fourier series of period with at least three periods.

In the graph 1 , by Dirichlet condition

At x=0,

$f_c\left(0\right)=0$

At x=1 ,

$f_c\left(1\right)=-\frac{1}{2}$

At x=2,

$f_c\left(2\right)=-2$

At x=-2 ,

$f_c\left(-2\right)=-2$

So,$f_c\left(0\right)=0,f_c\left(1\right)=-\frac{1}{2},f_c\left(2\right)=-1,f_c\left(-2\right)=-2$

In the graph 2 , by Dirichlet condition

At x=0 ,

$f_e\left(0\right)=-1$

At x=1 ,

$f_e\left(1\right)=-\frac{1}{2}$

At x=$\frac{3}{2}$ ,

$f_e\left(\frac{3}{2}\right)=-2$

At x=-2 ,

$f_e\left(-2\right)=-1$

So,$f_e\left(0\right)=-1,f_e\left(1\right)=-\frac{1}{2},f_e\left(\frac{3}{2}\right)=-2,f_e\left(-2\right)=-1$