(cos(A+B)+i sin(A+B))= exp i(A+B)=exp iA exp iB = (cosA+i sinA)(cosB+i sinB)= cosA cosB-sinAsinB +i(cosAsinB+sinAcosB) And: z'=z exp(ia)= (x+iy)(cos(a) + isin(a))= xcos(a)-ysin (a) +i(xsin(a)+ycos(a)) x'+iy'
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So multiplying a complex number (z=x+iy) by i=sqrt(-1) equals a 90 degrees rotation of the vector (x,y). z'=exp(i pi/2)z=iz, x'+iy'=-y+ix so x'=-y and y'=x.
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