We present a new and simple proof for the Hankel inversion formula. This formula was claimed by Hankel in 1869 and given many proofs later. Our result generalizes similar results by Zemanian (1968) and Duran (1990) to complex order Hankel transforms. The proof is based on representation theoretic ideas and will allow us to completely describe the Kirillov model for discrete series representations of $SL(2,R) and GL(2,R)$.