Relation between sine and cosine (Fourier transform)

Relation b/w fs(u) and fc(u) in fourier series soln: this is the method of the fourier series the topice is the fourier tansformation In fourier transformation the formula is F(x)=a0/2+∑fc(u)cosux+∑fs(u)sinux Fs(u)=∫_0^∞▒〖f(x)sinux.dx〗 Fc(u)=∫_0^∞▒〖f(x)cosux.dx〗 Now the relation b/w fs(u) and fc(u) is Fs(u)=∫_0^∞▒〖f(x)sinux.dx〗 d fs(u)/du=∫_0^∞▒〖f(x)cosux.dx〗

              =x∫_0^∞▒〖f(x)cosux.dx-∬_0^∞▒〖(f(x)cosux.dx)dx〗〗                         

d fs(u)/du=xfc(u) -∫_0^∞▒〖f(u)dx〗 here f(u) indicates fc(u) now the integrating with respect to u both side

           =∫_0^∞▒〖x f(u).du-〗  ∬_0^∞▒〖f(u).dx.du〗

Fs(u)=x∫_0^∞▒〖f(u).du-∬_0^∞▒(f(u).du)dx〗 Simplification Let β=∫_0^∞▒〖f(u)).du〗 F(u)=xβ-∫_0^∞▒〖β.dx〗