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Ministry of Higher Education and Scientifics research University of Babylon Complex Functions Lecture 11 Ali Hussein Mahmood Al-Obaidi College of Education for Pure Sciences Physics Department Three Stage ali.alobaidi81@yahoo.com 2.1 Function of a complex variables: Complex Variable:- A symbol , such as , which can stand for any one of a set complex numbers is called a complex variable. is called domain of variation. Complex function:- If to each value of a complex variable which can assume there corresponds one or more values of a complex variable , we say that is a function of and write or . The variable is sometimes called an independent variable, while is called a dependent variable. The value of a function at is often written . Thus if , then . 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 , we say that is a single-valued function. If more than one value of corresponds to each value of , we say that is a multiple-valued or many-valued function of . Ex:1) The function is single-valued function. 2) The function is multiple-valued function because 1 Ministry of Higher Education and Scientifics research University of Babylon Complex Functions Lecture 11 College of Education for Pure Sciences Physics Department Three Stage Ali Hussein Mahmood Al-Obaidi ali.alobaidi81@yahoo.com Inverse function:- If then we can also consider as a function of , written . The function is often called the inverse function corresponding to Thus and one inverse functions of each other. EX:- Find inverse function corresponding to Sol:Thus , where where , where One-one function:- Let called one-one function if . is a function, then a function is 2.2 Transformations:If function of (where and are real function) is a single-valued (where and are real numbers). we can write . By equating real and imaginary parts this is seen to be equivalent to , ………. Thus given a point in the plan , such as in (fig.1) below , there corresponds a point ( in the plane ,say in (fig.2) below .The set of equations [or the equivalent , ]is called a transformation. We say that point is transformed Q P into point by means of the Transformed and call the image of . Fig.1 2 Fig.2 Ministry of Higher Education and Scientifics research University of Babylon Complex Functions Lecture 11 Ali Hussein Mahmood Al-Obaidi Ex:- If College of Education for Pure Sciences Physics Department Three Stage ali.alobaidi81@yahoo.com then Then the transformation is . The image of a point (1,2) in the plane is the point (-3,4) in the plane . In general, under a transformation, a set of points such as these on curve of (fig.1) is mapped into corresponding set of points , called the image , such as those on curve in(fig.2). The particular characteristic of the image depend of course on the type of function , which is sometimes called a mapping function. If is multiplevalued, a point (or curve) in the plane is mapping in general in to more than one point (or curve) in the plane. 3