RAYAT SHIKSHAN SANSTHA’S S.M.JOSHI COLLEGE HADAPSAR, PUNE-411028

1. RAYAT SHIKSHAN SANSTHA’SS.M.JOSHI COLLEGE HADAPSAR, PUNE-411028 PRESANTATION BY Prof . DESAI S.S Mathematics department Subject – Complex Analysis Topic - Basics Definations

2. Open Disks or Neighborhoods • Definition. The set of all points z which satisfy the inequality |z – z0|<, where  is a positive real number is called an open disk or neighborhood of z0 . • Remark. The unit disk, i.e., the neighborhood |z|< 1, is of particular significance. 1

3. Interior Point • Definition. A point is called an interior point of S if and only if there exists at least one neighborhood of z0which is completely contained in S. z0 S

4. Open Set. Closed Set. • Definition. If every point of a set S is an interior point of S, we say that S is an open set. • Definition. If B(S)  S, i.e., if S contains all of its boundary points, then it is called a closed set. • Sets may be neither open nor closed. Neither Open Closed

5. Connected • An open set S is said to be connected if every pair of points z1and z2in S can be joined by a polygonal line that lies entirely in S. Roughly speaking, this means that S consists of a “single piece”, although it may contain holes. S z1 z2

6. Domain, Region, Closure, Bounded, Compact • An open, connected set is called a domain. A region is a domain together with some, none, or all of its boundary points. The closure of a set S denoted , is the set of S together with all of its boundary. Thus . • A set of points S is bounded if there exists a positive real number R such that |z|<R for every z  S. • A region which is both closed and bounded is said to be compact.

7. Review: Real Functions of Real Variables • Definition. Let   . A function f is a rule which assigns to each element a   one and only one element b  ,   . We write f:  , or in the specific case b = f(a), and call b “the image of a under f.” We call  “the domain of definition of f ” or simply “the domain of f ”. We call  “the range of f.” We call the set of all the images of , denoted f (), the image of the function f . We alternately call f a mapping from  to .

8. Real Function • In effect, a function of a real variable maps from one real line to another. f  

9. Complex Function • Definition. Complex function of a complex variable. Let   C. A function f defined on  is a rule which assigns to each z   a complex number w. The number w is called a value of f at z and is denoted by f(z), i.e., w = f(z). The set  is called the domain of definition of f. Although the domain of definition is often a domain, it need not be.

10. Remark • Properties of a real-valued function of a real variable are often exhibited by the graph of the function. But when w = f(z), where z and w are complex, no such convenient graphical representation is available because each of the numbers z and w is located in a plane rather than a line. • We can display some information about the function by indicating pairs of corresponding points z = (x,y) and w = (u,v). To do this, it is usually easiest to draw the z and w planes separately.

11. THANK YOU