Fourier Transform Basics-1

 $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ dx\ =\ F\left[s\right]$  

 $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)e^{isx}\ dx\ =\ F\left[s\right]$  

 $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ ds\ =\ F\left[s\right]$  

 $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ \ \cos sx\ dx\ =\ F\left[s\right]$  

 $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\ F\left[s\right]\ dx\ =\ f\left(x\right)$  

 $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}F\left[s\right]e^{isx}\ ds\ =f\left(x\right)$  

 $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}F\left[s\right]\ e^{-isx}ds\ =f\left(x\right)$  

 $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}F\left[s\right]\ \ \cos sx\ ds\ =\ f\left(x\right)$  

 $e^{isx\ }\ =\ \cos sx\ +\ \sin sx\ \ and\ e^{-isx\ }\ =\ \cos sx\ -\ \sin sx\ \$  

 $e^{isx\ }\ =\ \cos sx\ +i\sin sx\ \ and\ e^{-isx\ }\ =\ \cos sx\ -i\sin sx\ \$  

 $e^{isx\ }\ =\ \cos sx\ \ and\ e^{-isx\ }\ =\ \ \sin sx\ \$  

None of the above

sinsx

i sinsx

cossx

cosx

sinsx

i sinsx

cossx

cosx

 $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ dx\ =\ F_s\left[s\right]$  

 $\frac{1}{\sqrt{2\pi}}\int_0^{\infty}f\left(x\right)\ \sin sx\ dx\ =\ F_s\left[s\right]$  

 $\sqrt{\frac{2}{\pi}}\int_0^{\infty}f\left(x\right)\ \sin sx\ dx\ =\ F_s\left[s\right]$  

 $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ \ \cos sx\ dx\ =\ F_s\left[s\right]$  

 $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ ds=\ F_s\left[s\right]$  

 $\frac{1}{\sqrt{2\pi}}\int_0^{\infty}\ F_s\left[s\right]\sin sx\ ds=f\left(x\right)\$  

 $\sqrt{\frac{2}{\pi}}\int_0^{\infty}F_s\left[s\right]\sin sx\ ds\ =\ f\left(x\right)\$  

 $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\ F_s\left[s\right]\ \ \cos sx\ dx\ =f\left(x\right)$  

 $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ \cos sx\ ds=\ F_c\left[s\right]$  

 $\sqrt{\frac{2}{\pi}}\int_0^{\infty}\ F_c\left[s\right]\ \cos sx\ \ ds=f\left(x\right)\$  

 $\sqrt{\frac{2}{\pi}}\int_0^{\infty}F_s\left[s\right]\sin sx\ ds\ =\ f\left(x\right)\$  

 $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\ F_s\left[s\right]\ \ \cos sx\ dx\ =f\left(x\right)$  

 $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ \cos sx\ dx\ =\ F_c\left[s\right]$  

 $\frac{1}{\sqrt{2\pi}}\int_0^{\infty}f\left(x\right)\ \cos sx\ dx\ =\ F_c\left[s\right]$  

 $\sqrt{\frac{2}{\pi}}\int_0^{\infty}f\left(x\right)\ \cos sx\ dx\ =\ F_c\left[s\right]$  

 $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f\left(x\right)\ \ \cos sx\ dx\ =\ F_s\left[s\right]$  

complex form

Complex pair

complex transform pair

none of the above