Seonglae Cho2017143020 조성래
3.43
(a). (i).
Because of Euler’s formula, for odd is and for even that is zero because of odd harmonic ∴
(a). (ii).
It becomes zero when k is odd because of part is odd harmonic
(b). so
(c). We can’t prove (a).(i) part if it is even harmonic. and so fundamental period of is smaller than
(d). (1). Since the period of basic element is , the fundamental period is
(d). (2). In the equation, the period of element is so if there is no common factor, the lease common multiple is
3.44
Since is real, , and is real number ∴
because . means . This can’t be occur so is zero because is a positive real number.
means which means .
Finally, ()
3.45
(a).
Because cos is even, sin is odd function.
(b). ,
(c). 2
3.49
(a). for even , like below equation.
(b). like (a), for integer
(c). if is integer, for for integer
3.51
Since , for natural number . Also by simple computation, for natural number . ∴ and
3.53
(a). If is even, and are real if is real like below
(b). If is odd, is still real
3.56
(a). Using the multiplication property and
(b). Since is the sum of multiplication of , is also real