Complex Analysis- Lecture 11 - Function of a complex variables

2.1 Function of a complex variables:-
Complex Variable:- A symbol , such as Z , which can stand for any one of a set complex numbers D is called a complex variable. D is called domain of variation.
Complex function:- If to each value of a complex variable Z which can assume there corresponds one or more values of a complex variable ? , we say that ? is a function of Z and write ?=f(Z) or ?=G(Z). The variable Z is sometimes called an independent variable, while ? is called a dependent variable. The value of a function at Z=a is often written f(a). Thus if f(Z)=Z^2-4, then f(2i)=?(2i)?^2-4. D is called domain of the function and the set of value of a function ? is called range of the function.
Single and Multiple-valued function:- If only one value of ? corresponds to each value of Z , we say that ? is a single-valued function. If more than one value of ? corresponds to each value of Z, we say that ? is a multiple-valued or many-valued function of Z .
Ex:-
The function ?=Z^2+Z+1 is single-valued function.
The function ?=Z^(1/3) is multiple-valued function because
?=Z^(1/3)=r^(1/3) (cos??(?+2?k)/3?+i sin??(?+2?k)/3? ),k=0,1,2
Inverse function:- If ?=f(Z), then we can also consider Z as a function of ?, written Z=g(?)=f^(-1) (?). The function f^(-1) is often called the inverse function corresponding to f. Thus ?=f(Z) and z=f^(-1) (?) one inverse functions of each other.

EX:- Find inverse function corresponding to ?=f(Z)=1/(Z+2) , where Z?-2.
Sol:- ?=1/(Z+2)?z+2=1/? ?z=1/?-2, where ??0
Thus Z=f^(-1) (?)=1/?-2 , where ??0 .
One-one function:- Let ?=f(Z) is a function, then a function ? is called one-one function if
f(Z_1 )=f(Z_2 )?Z_1=Z_2
2.2 Transformations:-
If ?=u+iv (where u and v are real function) is a single-valued function of Z=x+iy (where x and y are real numbers). we can write u+iv=f(x+iy). By equating real and imaginary parts this is seen to be equivalent to u=u(x,y) , v=v(x,y). ……….*
Thus given a point (x,y) in the Z plan , such as P in (fig.1) below , there corresponds a point (u,v) in the ? plane ,say P_1 in (fig.2) below .The set of equations [or the equivalent ,
?=f(Z)]is called a transformation.
We say that point P is transformed
into point P_1 by means of the
Transformed and call P_1 the image
of P.



Ex:- If ?= Z^2 then
u+iv=(x+iy)^2=x^2+2xyi+i^2 y^2=(x^2-y^2 )+2xyi
Then the transformation is u=x^2-y^2 &v=2xy. The image of a point (1,2) in the plane Z is the point (-3,4) in the ?plane .
In general, under a transformation, a set of points such as these on curve PQ of (fig.1) is mapped into corresponding set of points , called the image , such as those on curve P_1 Q_1 in(fig.2). The particular characteristic of the image depend of course on the type of function f(Z), which is sometimes called a mapping function. If f(Z) is multiple-valued, a point (or curve) in the Z plane is mapping in general in to more than one point (or curve) in the ? plane.