Differentiating Under the Integral Sign
Let R be a region and let K be a complex valued function of two variables z in R and t in [a,b] such 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 t in [a,b].
Hence, if f is given by (1) then
say. We need to show thatas
henceas