Proving the Differentiation Under the Integral Sign Formula
Theorem
Letbe a region and letbe a complex valued function of two variablesandsuch that

is analytic inas a function offor each

andare continuous onas functions of t for each

For somefor
Then the functionwithis analytic onandfor(1)
Proof: Letand choose a circleinwith centreand radiussuch that the inside oflies entirely inIflies insidethen we have by assumption 1 and Cauchy's Integral Formula,
and
and by Cauchy's First Derivative Formula,for each
Hence, if f is given by (1) then
say. We need to show thatas
henceas