Dierentiating Under The Integral Sign
Let f (z, ) be a function of two complex variables z and . Let be a piecewise smooth path
in C. Consider,
f (z, )d.
We will want a criterion that allows us to conclude that
#1: Find the image of cfw_z: Re z < 0, |Im z| < p under the exponential function.
ez = eu(x, y) + iv(x, y) = eueiv. The real part of z must be less than zero, hence we have
Math 210A Homework 4
Exercise 1. Express the functions ez , cos z, sin z, (1 + z) , log(1 + z), and
their radii of convergence.
as a power series in z. Find
1 + z2
The radius of convergence is:
R = lim
THE CONCEPT OF
AN ANALYTIC FUNCTION
The theory of functions is concerned with complex-valued functions of a
complex variable. Our study is conﬁned to those functions which are
dz'ﬂerentiable in a sense which will be made precise later on; such f
BULLETIN (New Series) OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 6, Number 2, March 1982
THE CAUCHY-RIEMANN EQUATIONS
AND DIFFERENTIAL GEOMETRY
BY R. O. WELLS, JR.
1. Introduction. In 1907 Poincar wrote a seminal paper  on various
topics in several c