Analyticity

A complex number has the form

$$\fbox {$\displaystyle {z=x+y\textrm{i}}$}$$

where x and y are real numbers and i²=−1. We write ℜ(z):=x and ℑ(z):=y and call this the real and imaginary part of z, respectively. The set of all complex numbers is denoted by ℂ; it is also called the complex plane. Using polar coordinates, each complex number z=x+yi can be written uniquely as

$$x=r\cos \varphi,\qquad y=r\sin \varphi,\qquad -\pi <\varphi \le \pi $$

where $r:=\sqrt{x^{2}+y^{2}}.$ The real numbers

$$|z|:=r,\qquad \arg z:=\varphi $$

are called the modulus and the principal argument of z, respectively (Fig. 4.1). Using the Euler formula e^iφ=cos φ+isin φ, each complex number z can also be uniquely represented as

$$z=|z|\cdot \textrm{e}^{\textrm{i}\varphi},\qquad -\pi <\varphi \le \pi.$$

Sometimes it is useful to use the representation

$$z=|z|\textrm{e}^{\textrm{i}\varphi_*}$$

where φ_*=arg (z)+2πk with k=0,±1,±2,… We call φ_* an argument of the complex number, and we write arg _*(z)=φ₊.