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Basic concept of Complex Function

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### Complex function

1. 1. Complex Function N. B. Vyas Department of Mathematics, Atmiya Institute of Tech. and Science, Rajkot (Guj.)N.B.V yas − Department of M athematics, AIT S − Rajkot (2)
2. 2. Curves and Regions in Complex PlaneDistance between two complex numbersThe distance between two complex numbers z1 and z2 is given by|z1 − z2 | or |z2 − z1 | N.B.V yas − Department of M athematics, AIT S − Rajkot (3)
3. 3. Curves and Regions in Complex PlaneDistance between two complex numbersThe distance between two complex numbers z1 and z2 is given by|z1 − z2 | or |z2 − z1 |CirclesA circle with centre z0 = (x0 , y0 ) C and radius p R+ isrepresented by |z − z0 | = p N.B.V yas − Department of M athematics, AIT S − Rajkot (3)
4. 4. Curves and Regions in Complex PlaneDistance between two complex numbersThe distance between two complex numbers z1 and z2 is given by|z1 − z2 | or |z2 − z1 |CirclesA circle with centre z0 = (x0 , y0 ) C and radius p R+ isrepresented by |z − z0 | = pInterior and exterior part of the circle |z − z0 | = pThe set {z C, p R+ /|z − z0 | < p} indicates the interior part ofthe circle |z − z0 | = p N.B.V yas − Department of M athematics, AIT S − Rajkot (3)
5. 5. Curves and Regions in Complex PlaneDistance between two complex numbersThe distance between two complex numbers z1 and z2 is given by|z1 − z2 | or |z2 − z1 |CirclesA circle with centre z0 = (x0 , y0 ) C and radius p R+ isrepresented by |z − z0 | = pInterior and exterior part of the circle |z − z0 | = pThe set {z C, p R+ /|z − z0 | < p} indicates the interior part ofthe circle |z − z0 | = p whereas {z C, p R+ /|z − z0 | > p} indicatesexterior part of it. N.B.V yas − Department of M athematics, AIT S − Rajkot (4)
6. 6. Curves and Regions in Complex PlaneCircular DiskThe open circular disk with centre z0 and radius p is given byz C, p R+ /|z − z0 | < p. N.B.V yas − Department of M athematics, AIT S − Rajkot (5)
7. 7. Curves and Regions in Complex PlaneCircular DiskThe open circular disk with centre z0 and radius p is given byz C, p R+ /|z − z0 | < p. The close circular disk is given by{z C, p R+ /|z − z0 | ≤ p} N.B.V yas − Department of M athematics, AIT S − Rajkot (5)
8. 8. Curves and Regions in Complex PlaneCircular DiskThe open circular disk with centre z0 and radius p is given byz C, p R+ /|z − z0 | < p. The close circular disk is given by{z C, p R+ /|z − z0 | ≤ p}NeighbourhoodAn open neighbourhood of a point z0 is a subset of Ccontaining an open circular disk centered at z0 . N.B.V yas − Department of M athematics, AIT S − Rajkot (5)
9. 9. Curves and Regions in Complex PlaneCircular DiskThe open circular disk with centre z0 and radius p is given byz C, p R+ /|z − z0 | < p. The close circular disk is given by{z C, p R+ /|z − z0 | ≤ p}NeighbourhoodAn open neighbourhood of a point z0 is a subset of Ccontaining an open circular disk centered at z0 . MathematicallyNp (z0 ) = {z C, p R+ /|z − z0 | < p} N.B.V yas − Department of M athematics, AIT S − Rajkot (5)
10. 10. Curves and Regions in Complex PlaneCircular DiskThe open circular disk with centre z0 and radius p is given byz C, p R+ /|z − z0 | < p. The close circular disk is given by{z C, p R+ /|z − z0 | ≤ p}NeighbourhoodAn open neighbourhood of a point z0 is a subset of Ccontaining an open circular disk centered at z0 . MathematicallyNp (z0 ) = {z C, p R+ /|z − z0 | < p}A punctured or deleted neighbourhood of a point z0 containall the points of a neighbourhood of z0 , excepted z0 itself.Mathematically {z C, p R+ /0 < |z − z0 | < p} N.B.V yas − Department of M athematics, AIT S − Rajkot (6)
11. 11. Curves and Regions in Complex PlaneAnnulusThe region between two concentric circles with centre z0 of radiip1 and p2 (> p1 ) can be represented by p1 < |z − z0 | < p2 . N.B.V yas − Department of M athematics, AIT S − Rajkot (7)
12. 12. Curves and Regions in Complex PlaneAnnulusThe region between two concentric circles with centre z0 of radiip1 and p2 (> p1 ) can be represented by p1 < |z − z0 | < p2 . Such aregion is called open circular ring or open annulus. N.B.V yas − Department of M athematics, AIT S − Rajkot (7)
13. 13. Curves and Regions in Complex PlaneAnnulusThe region between two concentric circles with centre z0 of radiip1 and p2 (> p1 ) can be represented by p1 < |z − z0 | < p2 . Such aregion is called open circular ring or open annulus. N.B.V yas − Department of M athematics, AIT S − Rajkot (8)
14. 14. Curves and Regions in Complex PlaneOpen SetLet S be a subset of C. It is called an open set if for eachpoints z0 S, there exists an open circular disk centered at z0which included in S. N.B.V yas − Department of M athematics, AIT S − Rajkot (9)
15. 15. Curves and Regions in Complex PlaneOpen SetLet S be a subset of C. It is called an open set if for eachpoints z0 S, there exists an open circular disk centered at z0which included in S.Closed SetA set S is called closed if its complement is open. N.B.V yas − Department of M athematics, AIT S − Rajkot (9)
16. 16. Curves and Regions in Complex PlaneOpen SetLet S be a subset of C. It is called an open set if for eachpoints z0 S, there exists an open circular disk centered at z0which included in S.Closed SetA set S is called closed if its complement is open.Connected SetA set A is said to be connected if any two points of A can bejoined by ﬁnitely many line segments such that each point on theline segment is a point of A N.B.V yas − Department of M athematics, AIT S − Rajkot (10)
17. 17. Curves and Regions in Complex PlaneDomainAn open connected set is called a domain. N.B.V yas − Department of M athematics, AIT S − Rajkot (11)
18. 18. Curves and Regions in Complex PlaneDomainAn open connected set is called a domain.RegionIt is a domain with some of its boundary points. N.B.V yas − Department of M athematics, AIT S − Rajkot (11)
19. 19. Curves and Regions in Complex PlaneDomainAn open connected set is called a domain.RegionIt is a domain with some of its boundary points.Closed regionIt is a region together with the boundary points (all boundarypoints included). N.B.V yas − Department of M athematics, AIT S − Rajkot (11)
20. 20. Curves and Regions in Complex PlaneDomainAn open connected set is called a domain.RegionIt is a domain with some of its boundary points.Closed regionIt is a region together with the boundary points (all boundarypoints included).Bounded regionA region is said to be bounded if it can be enclosed in a circleof ﬁnite radius. N.B.V yas − Department of M athematics, AIT S − Rajkot (12)
21. 21. Function of a Complex Variable If z = x + iy and w = u + iw are two complex variables and if to each point z of region R there corresponds at least on point w of a region R we say that w is a function of z and we write w = f (z) N.B.V yas − Department of M athematics, AIT S − Rajkot (13)
22. 22. Function of a Complex Variable If z = x + iy and w = u + iw are two complex variables and if to each point z of region R there corresponds at least on point w of a region R we say that w is a function of z and we write w = f (z) If for each value of z in a region R of the z-plane there corresponds a unique value for w then w is called single valued function. N.B.V yas − Department of M athematics, AIT S − Rajkot (13)
23. 23. Function of a Complex Variable If z = x + iy and w = u + iw are two complex variables and if to each point z of region R there corresponds at least on point w of a region R we say that w is a function of z and we write w = f (z) If for each value of z in a region R of the z-plane there corresponds a unique value for w then w is called single valued function.E.g.: w = z 2 is a single valued function of z. N.B.V yas − Department of M athematics, AIT S − Rajkot (13)
24. 24. Function of a Complex Variable If z = x + iy and w = u + iw are two complex variables and if to each point z of region R there corresponds at least on point w of a region R we say that w is a function of z and we write w = f (z) If for each value of z in a region R of the z-plane there corresponds a unique value for w then w is called single valued function.E.g.: w = z 2 is a single valued function of z. If for each value of z if more than one value of w exists then w is called multi-valued function. N.B.V yas − Department of M athematics, AIT S − Rajkot (13)
25. 25. Function of a Complex Variable If z = x + iy and w = u + iw are two complex variables and if to each point z of region R there corresponds at least on point w of a region R we say that w is a function of z and we write w = f (z) If for each value of z in a region R of the z-plane there corresponds a unique value for w then w is called single valued function.E.g.: w = z 2 is a single valued function of z. If for each value of z if more than one value of w exists then w is called multi-valued function. √E.g.: w = Z N.B.V yas − Department of M athematics, AIT S − Rajkot (13)
26. 26. Function of a Complex Variable If z = x + iy and w = u + iw are two complex variables and if to each point z of region R there corresponds at least on point w of a region R we say that w is a function of z and we write w = f (z) If for each value of z in a region R of the z-plane there corresponds a unique value for w then w is called single valued function.E.g.: w = z 2 is a single valued function of z. If for each value of z if more than one value of w exists then w is called multi-valued function. √E.g.: w = Z w = f (z) = u(x, y) + iv(x, y) where u(x, y) and v(x, y) are known as real and imaginary parts of the function w. N.B.V yas − Department of M athematics, AIT S − Rajkot (13)
27. 27. Function of a Complex Variable If z = x + iy and w = u + iw are two complex variables and if to each point z of region R there corresponds at least on point w of a region R we say that w is a function of z and we write w = f (z) If for each value of z in a region R of the z-plane there corresponds a unique value for w then w is called single valued function.E.g.: w = z 2 is a single valued function of z. If for each value of z if more than one value of w exists then w is called multi-valued function. √E.g.: w = Z w = f (z) = u(x, y) + iv(x, y) where u(x, y) and v(x, y) are known as real and imaginary parts of the function w.E.g.: f (z) = z 2 = (x + iy)2 = (x2 − y 2 ) + i(2xy) N.B.V yas − Department of M athematics, AIT S − Rajkot (13)
28. 28. Function of a Complex Variable If z = x + iy and w = u + iw are two complex variables and if to each point z of region R there corresponds at least on point w of a region R we say that w is a function of z and we write w = f (z) If for each value of z in a region R of the z-plane there corresponds a unique value for w then w is called single valued function.E.g.: w = z 2 is a single valued function of z. If for each value of z if more than one value of w exists then w is called multi-valued function. √E.g.: w = Z w = f (z) = u(x, y) + iv(x, y) where u(x, y) and v(x, y) are known as real and imaginary parts of the function w.E.g.: f (z) = z 2 = (x + iy)2 = (x2 − y 2 ) + i(2xy) ∴ u(x, y) = x2 − y 2 and v(x, y) = 2xy N.B.V yas − Department of M athematics, AIT S − Rajkot (14)
29. 29. Limit and Continuity of f (z) A function w = f (z) is said to have the limit l as z approaches a point z0 if for given small positive number ε we can ﬁnd positive number δ such that for all z = z0 in a disk |z − z0 | < δ we have |f (z) − l| < ε N.B.V yas − Department of M athematics, AIT S − Rajkot (15)
30. 30. Limit and Continuity of f (z) A function w = f (z) is said to have the limit l as z approaches a point z0 if for given small positive number ε we can ﬁnd positive number δ such that for all z = z0 in a disk |z − z0 | < δ we have |f (z) − l| < ε Symbolically, we write lim f (z) = l z→z0 N.B.V yas − Department of M athematics, AIT S − Rajkot (15)
31. 31. Limit and Continuity of f (z) A function w = f (z) is said to have the limit l as z approaches a point z0 if for given small positive number ε we can ﬁnd positive number δ such that for all z = z0 in a disk |z − z0 | < δ we have |f (z) − l| < ε Symbolically, we write lim f (z) = l z→z0 A function w = f (z) = u(x, y) + iv(x, y) is said to be continuous at z = z0 if f (z0 ) is deﬁned and lim f (z) = f (z0 ) z→z0 N.B.V yas − Department of M athematics, AIT S − Rajkot (15)
32. 32. Limit and Continuity of f (z) A function w = f (z) is said to have the limit l as z approaches a point z0 if for given small positive number ε we can ﬁnd positive number δ such that for all z = z0 in a disk |z − z0 | < δ we have |f (z) − l| < ε Symbolically, we write lim f (z) = l z→z0 A function w = f (z) = u(x, y) + iv(x, y) is said to be continuous at z = z0 if f (z0 ) is deﬁned and lim f (z) = f (z0 ) z→z0 In other words if w = f (z) = u(x, y) + iv(x, y) is continuous at z = z0 then u(x, y) and v(x, y) both are continuous at (x0 , y0 ) N.B.V yas − Department of M athematics, AIT S − Rajkot (15)
33. 33. Limit and Continuity of f (z) A function w = f (z) is said to have the limit l as z approaches a point z0 if for given small positive number ε we can ﬁnd positive number δ such that for all z = z0 in a disk |z − z0 | < δ we have |f (z) − l| < ε Symbolically, we write lim f (z) = l z→z0 A function w = f (z) = u(x, y) + iv(x, y) is said to be continuous at z = z0 if f (z0 ) is deﬁned and lim f (z) = f (z0 ) z→z0 In other words if w = f (z) = u(x, y) + iv(x, y) is continuous at z = z0 then u(x, y) and v(x, y) both are continuous at (x0 , y0 ) And conversely if u(x, y) and v(x, y) both are continuous at (x0 , y0 ) then f (z) is continuous at z = z0 . N.B.V yas − Department of M athematics, AIT S − Rajkot (16)
34. 34. Differentiation of f (z) The derivative of a complex function w = f (z) a point z0 is written as f (z0 ) and is deﬁned by dw f (z0 + δz) − f (z0 ) = f (z0 ) = lim provided limit exists. dz δz→0 δz N.B.V yas − Department of M athematics, AIT S − Rajkot (17)
35. 35. Differentiation of f (z) The derivative of a complex function w = f (z) a point z0 is written as f (z0 ) and is deﬁned by dw f (z0 + δz) − f (z0 ) = f (z0 ) = lim provided limit exists. dz δz→0 δz Then f is said to be diﬀerentiable at z0 if we write the change δz = z − z0 since z = z0 + δz f (z) − f (z0 ) ∴ f (z0 ) = lim z→z0 z − z0 N.B.V yas − Department of M athematics, AIT S − Rajkot (18)
36. 36. Analytic Functions A single - valued complex function f (z) is said to be analytic at a point z0 in the domain D of the z−plane, if f (z) is diﬀerentiable at z0 and at every point in some neighbourhood of z0 . Point where function is not analytic (i.e. it is not single valued or not) are called singular points or singularities. From the deﬁnition of analytic function 1 To every point z of R, corresponds a deﬁnite value of f (z). 2 f (z) is continuous function of z in the region R. 3 At every point of z in R, f (z) has a unique derivative. N.B.V yas − Department of M athematics, AIT S − Rajkot (19)
37. 37. Cauchy-Riemann Equation f is analytic in domain D if and only if the ﬁrst partial derivative of u and v satisfy the two equations ∂u ∂v ∂u ∂v = , =− − − − − − (1) ∂x ∂y ∂y ∂x The equation (1) are called C-R equations. N.B.V yas − Department of M athematics, AIT S − Rajkot (20)
38. 38. ExampleEx. Find domain of the following functions: 1 1 2 z +1 1Sol. Here f (z) = 2 z +1 f (z) is undeﬁned if z = i and z = −i ∴ Domain is a complex plane except z = ±i 1 2 arg z 1Sol. Here f (z) = arg z 1 1 1 x − iy x − iy = = x = 2 z x + iy x + iy x − iy x + y2 1 ∴ is undeﬁned for z = 0 z 1 Domain of arg is a complex plane except z = 0. N.B.V yas −z Department of M athematics, AIT S − Rajkot (21)
39. 39. Example z 3 z+z¯ zSol. Here f (z) = z+z ¯ f (z) is undeﬁned if z + z = 0 ¯ i.e. (x + iy) + (x − iy) = 0 ∴ 2x = 0 ∴x=0 f (z) is undeﬁned if x = 0 Domain is complex plane except x = 0 N.B.V yas − Department of M athematics, AIT S − Rajkot (22)