698 CHAPTER 35. ANALYTIC FUNCTIONS

like long polynomials. In fact, this is the correct context for the study of power series.Then from calculus, the next thing considered is the rational functions. The generalizationof this simple concept will be the meromorphic functions. Remarkable things are obtainedfrom these simple considerations. Surprising applications are available when this theory isdeveloped. I will demonstrate that these extravagant assertions are abundantly verified.

We will be considering line integrals and it will be assumed that the curves over whichthe line integrals are taken are piecewise C1. Actually, all that is needed is that thesecurves have finite length but this is better considered in a book devoted primarily to themathematical theory.

35.2 The Cauchy Riemann EquationsThese fundamental equations pertain to a complex valued function of a complex variable.Recall the complex numbers should be considered as points in the plane. Thus a complexnumber is of the form x+ iy where i2 =−1. The complex conjugate is defined by

x+ iy≡ x− iy

and for z a complex number,

|z| ≡ (zz)1/2 =√

x2 + y2.

Thus when x+ iy is considered an ordered pair (x,y) ∈ R2 the magnitude of a complexnumber is nothing more than the usual norm of the ordered pair. Also for z = x+ iy,w =u+ iv,

|z−w|=√(x−u)2 +(y− v)2

so in terms of all topological considerations, R2 is the same as C. Thus to say z→ f (z) iscontinuous, is the same as saying

(x,y)→ u(x,y) , (x,y)→ v(x,y)

are continuous where f (z)≡ u(x,y)+ iv(x,y) with u and v being called the real and imag-inary parts of f . The only new thing is that writing an ordered pair (x,y) as x+ iy withthe convention i2 = −1 makes C into a field. You should verify that for z,w two complexnumbers, |zw|= |z| |w| . Also z+w = z+w.

Now here is the definition of what it means for a function to be analytic.

Definition 35.2.1 Let U be an open subset of C (R2) and let f : U → C be a function.Then f is said to be analytic on U if for every z ∈U,

lim∆z→0

f (z+∆z)− f (z)∆z

≡ f ′ (z)

exists and is a continuous function of z ∈U. For a function having values in C denote byu(x,y) the real part of f and v(x,y) the imaginary part. Both u and v have real values and

f (x+ iy)≡ f (z)≡ u(x,y)+ iv(x,y)