Complex variable methods are introduced to derive exact and closed expressions for the stress functions, torsional rigidities, and peripheral shearing stresses of certain isotropic cylinders under torsion. Numerical results in some special cases are presented in tabular and graphic form. The equations for the boundaries of the cross sections for these cylinders have the polar forms r 2=a 2(2 cos 2θ−1), a>0 (‖θ‖≤π/6), r 4=2n a 4/(n−1+cos 4θ)(n+1+cos 4θ), a>0 (‖θ‖≤π, 20 (m>2, 0<δ<π/2m, ‖θ‖≤δ).