  We present a set of lectures on topics of advanced calculus in one real and
complex variable with several new results and proofs on the subject, specially
with detailed proof-always missing in the literature - of the Cissoti
explicitly integral formula conformally representing a polygon onto a
disc.Besides we present-in the paper appendix-a new study embodied with a
mathematical physicist perspective,on the famous Riemann conjecture on the
zeros of the Zeta function, reducing its proof to a conjecture on the
positivity of a numerical series.
