Here is the formula derived in the video 
This question does not appear to be about electronics design within the scope defined in the help center.
Closed 3 years ago .In this question, he converted the sinusoidal voltage to exponential voltage. what's the formula used here ? This is not the same one used in this video Here is the formula derived in the video
\$\begingroup\$ I’m voting to close this question because this is the most basic mathematical expression in harmonic analysis: Euler's relation. This is basic math, not EE. \$\endgroup\$
Commented Feb 22, 2021 at 13:18\$\begingroup\$ I agree with @ahmedosama , they are not mathematically equivalent. Mathematically, sin x = (e^jx - e^-jx)/2j. What is going on, is that electrical engineers tend to ignore the fact that one needs to add or subtract the complex conjugate to get a real value (or take the Re part). \$\endgroup\$
Commented Feb 22, 2021 at 14:40\$\begingroup\$ I think they are phase shifting the Euler formula 90 degrees with the J at the front since the real part of Euler is given in terms of cosine but your source function is given in sin. A sin is a 90 degree phase shifted cosine. More obvious if you change Euler to complex cartesian form and multiply by J. And they omitted taking the real part. It's sloppy. \$\endgroup\$
Commented Feb 22, 2021 at 15:12\$\begingroup\$ By multiplying by J and not -J they also phase shifted it 90 degrees in the wrong direction to get the real parts strictly equal, but since it is the only source it doesn't matter since phases only have meaning relative to each other, so long as you shift all phases by the same amount. As written the real parts are actually negatives of each other and not even equal. Real sloppy. In light of all this sloppiness and omissions I vote to reopen. \$\endgroup\$
