Complex Functions Limit and Continuity

1. Complex FunctionsLimit and Continuity Mohammed Nasser Acknowledgement: Steve Cunningham

2. Relation between MM (ML) and Vector space

3. Mathematical Concepts

4. Mathematical Concepts Covariance Variance

5. Z

8. Basis F 0 Set of f1, f2, …, fn is linearly independent if α1 f1+ α2 f2 + …+ αn fn = 0 ,Holds only if each αi=0. Finite dimensional if there exist maximum n linearly independent elements ; Otherwise it is infinite-dimensional Basis: can express every f in F in the form f = α1 f1+ α2 f2 + …+ αn fn Linear manifold: αf1+βf2 in F

9. Basis ( Continued)

10. Mathematical Concepts Covariance Variance

11. Length of a vector Right-angle triangle Pythagoras’ theorem || x || = (x12+ x22 + x32 )1/2 Inner product of a vector with itself = (vector length)2 xTx =x12+ x22 +x32 = (|| x ||)2 x2 ||x|| x1 Some Vector Concepts • Dot product = scalar x2 || x || = (x12+ x22)1/2 x1

12. ||x|| ||y|| b y2  q y1 x =/2 Orthogonal vectors: xT y = 0 y Some Vector Concepts • Angle between two vectors x

13. Dot Product inRn • XTy=yTx (aX+bZ)Ty=aXTY+bZTY • XTX=0 ↔ X=0

14. Real Inner product space Inner product space: A vector space X over the reals R is an inner product space if and only if there exists a real-valued symmetric bilinear (linear in each argument) map (.,.), that satisfies: < , >: X X→R <x,y>=<y,x> <x,x>.≥0, and <x, x>=0↔ x=0 <x+z,y>=<x,y>+<z,y> <kx,y>=k<x,y> Show that R is itself an inner product space with inner product <x1,x2>=x1 x2

15. Complex Inner product space Inner product space: A vector space X over the reals C is an inner product space if and only if there exists a real-valued symmetric bilinear (linear in each argument) map (.,.), that satisfies: < , >: X X→C <x,y>=<y,x> <x,x>.≥0, and <x, x>=0↔ x=0 <x+z,y>=<x,y>+<z,y> <kx,y>=k<x,y> Show that C/Z is itself an inner product space with inner product <z1,z2>=z1 z2

16. Inner product Space A vector space V on which an inner product is defined is called an inner product space.Any function on a vector space that satisfies the axioms of an inner product defines an inner product on the space. . There can be many inner products on a given vector space

17. Example 2 Let u = (x1, x2), v = (y1, y2), and w = (z1, z2) be arbitrary vectors in R2. Prove that<u, v>, defined as follows, is an inner product on R2. <u, v>= x1y1 + 4x2y2 Determine the inner product of the vectors (-2, 5), (3, 1) under this inner product. Solution Axiom 1:<u, v>= x1y1 + 4x2y2 = y1x1 + 4y2x2 =<v, u> Axiom 2:<u + v, w>=< (x1, x2) + (y1, y2) , (z1, z2) > =< (x1 + y1, x2 + y2), (z1, z2) . = (x1 + y1) z1 + 4(x2 + y2)z2 = x1z1 + 4x2z2 + y1 z1 + 4 y2z2 =<(x1, x2), (z1, z2)>+<(y1, y2), (z1, z2) > =<u, w>+<v, w>

18. Axiom 4: <u, u>= <(x1, x2), (x1, x2)>= Further, if and only if x1 = 0 and x2 = 0. That is u = 0. Thus<u, u> 0, and<u, u>= 0 if and only if u = 0. The four inner product axioms are satisfied, <u, v>= x1y1 + 4x2y2 is an inner product on R2. Axiom 3:<cu, v>= <c(x1, x2), (y1, y2)> =< (cx1, cx2), (y1, y2) > = cx1y1 + 4cx2y2 = c(x1y1 + 4x2y2) = c<u, v> The inner product of the vectors (-2, 5), (3, 1) is <(-2, 5), (3, 1)>= (-2  3) + 4(5  1) = 14

19. Example Consider the vector space M22 of 2  2 matrices. Let u and v defined as follows be arbitrary 2  2 matrices. Prove that the following function is an inner product on M22. <u, v>= ae + bf + cg + dh Determine the inner product of the matrices . Solution Axiom 1:,<u, v>= ae + bf + cg + dh = ea + fb + gc + hd =<v, u> Axiom 3: Let k be a scalar. Then <ku, v>= kae + kbf + kcg + kdh = k(ae + bf + cg + dh) = k<u, v>

20. Cauchy–Schwarz Inequality In an inner product space, |<x,y>|2≤ <x,x><y,y> and the equality sign holds in a strict inner product space if and only if x and y are rescalings of the same vector. Defining |x||=<x.x>1/2 we can make every inner product a normed space . Using CSI we can introduce concept of angle, orthogonality, correlation etc into any innerproduct space Projection theorem holds in this space

21. Angle between two vectors In R2 we first define cosθ, then prove C-S inequality In Rn we first prove C-S inequality , then define cosθ

22. Definition Let V be an inner product space. The angle between two nonzero vectors u and v in V is given by Angle between two vectors The dot product in Rn was used to define angle between vectors. The angle  between vectors u and v in Rn is defined by

23. Normed spaces Define the notion of the size of f, an element in F , a vector spacea vector space Norm || f ||, || ||: F→[0,∞) 1)||f||=0↔f=0 2) ||kf||=|k|||f|| 3) ||f|+||g|| <=||f||+||g|| Both R and Z are normed spaces are spaces with | | Both Rn and are normed spaces are spaces with Euclidean norm,|| ||

24. The norm of a vector in Rn can be expressed in terms of the dot product as follows Definition Let V be an inner product space. The norm of a vector v is denoted ||v|| and it defined by Norm of a Vector Generalize this definition: The norms in general vector space do not necessary have geometric interpretations, but are often important in numerical work.

25. Example Consider the vector space M22 of 2  2 matrices. Let u and v defined as follows be arbitrary 2  2 matrices. It is known that the function <u, v>= ae + bf + cg + dh is an inner product on M22 by Example 2. The norm of the matrix is

26. Different Norms on C/Z • We have already seen mod, | | is norm on C. • Let | |max and | |tax be two functions defined as follow: | z|max =max{|x|, |y|}, | z|tax =|x| +|y|, • Show that both are norms on C. • Show that | z|max =< |z|=<| z|tax =<2 | z|max In fact, it can be shown that for a finite-dimensional space norms are equivalent All norms are not generated from innerproduct.

27. Definition Let V be an inner product space with vector norm defined by The distance between two vectors (points) u and v is defined d(u,v) and is defined by Distance As for norm, the concept of distance will not have direct geometrical interpretation. It is however, useful in numerical mathematics to be able to discuss how far apart various functions are. Show that 1. || ||: F→[0,∞) is a continuous function. 2. d(f,g)=||f-g|| is a metric on F

28. Metric spaces Put some structure on our spaceF defining nonnegative function d:F χF→R f2 F f1 f3 1.d(f1,, f2)=d(f2,, f1), 2) d(f1,, f2)=0 if and only if f1=f2 3, d(f1,, f2)≤ d(f1,, f3) + d(f3,, f2)

29. Why are metric spaces important? Allow us to define the distance between functions Can be able to treat convergence in the space,and limit and continuity of metric space valued functions on metric space. Completeness – no holes in the space We want to look at spaces that are very similar to Euclidean space Can We Define Rate Of Change?? Can we talk about best approximations? Yes if Can we get the best from data?

30. Mathematical Concepts Covariance Variance

31. Sequence • Definition. A sequence of complex numbers, denoted , is a function f, such that f: N  C, i.e, it is a function whose domain is the set of natural numbers between 1 and k, and whose range is a subset of the complex numbers. If k = , then the sequence is called infinite and is denoted by , or more often, zn . (The notation f(n) is equivalent.) • Having defined sequences and a means for measuring the distance between points, we proceed to define the limit of a sequence.

32. Meaning of Zn Z0 Zn Z0 |zn-z0|=rn 0 Where rn= , xn x0,,, yn y0 I proved it in previous classes

33. Geometric Meaning of Zn Z0 zn tends to z0 in any linear or curvilinear way.

34. Limit of a Sequence • Definition. A sequence of complex numbers is said to have the limit z0 , or to converge to z0, if for any  > 0, there exists an integer N such that |zn – z0| <  for all n > N. We denote this by • Geometrically, this amounts to the fact that z0is the only point of znsuch that any neighborhood about it, no matter how small, contains an infinite number of points zn .

35. Geometric Meaning of Zn Z0 - - zN+2 z0 zN+1 zn tends to z0 in any linear or curvilinear way.

36. Example: Convergent Sequence • Given , choose N=1/ , p=0 1 0 Establish convergence by applying definition Necessitates knowledge of p.

37. Cauchy Sequence • A sequence in a metric space X such that for every , there is an integer N such that if • A sequence in a complex field C such that for every , there is an integer N such that if

38. Example Cauchy Sequence • Given , choose N=2/, p=0 1 0 Establish convergence by applying definition No need to know p

39. Theorems and Exercises Theorem. Show that zn=xn+iyn z0=x0+iy0 if and only if xn x0, yn y0 . Ex. Plot the first ten elements of the following sequences and find their limits if they exist: • 1/n +i 1/n • 1/n2 +i 1/n2 • n +i 1/n • (1-1/n )n +i (1+1/n)n

40. Topology Topology studies the invariant properties of object under continuous deformations e d For alle > 0, there existsd > 0 : |y-x| < d) |f(y)-f(x)| < e

41. Topology Topology studies the invariant properties of object under continuous deformations S f-1(S) For all S open, f-1(S) is open

42. Limit of a Function • We say that the complex number w0is the limit of the function f(z) as z approaches z0 if f(z) stays close to w0whenever z is sufficiently near z0 . Formally, we state: • Definition. Limit of a Complex Sequence. Let f(z) be a function defined in some neighborhood of z0 except with the possible exception of the point z0is the number w0if for any real number  > 0 there exists a positive real number  > 0 such that |f(z) – w0|<  whenever 0<|z - z0|< .

43. Limits: Interpretation We can interpret this to mean that if we observe points w within a radius  of w0, we can find a corresponding disk about z0such that all the points in the disk about z0are mapped into it. That is, any neighborhood of w0 contains all the values assumed by f in some full neighborhood of z0, except possibly f(z0). v y w = f(z)   w0 z0 u x w-plane z-plane

44. Complex Functions : Limit and Continuity f: Ω1 Ω2 Ω1 and Ω2 are domain and codomain respectively. Let z0 be a limit point of Ω1 , w0 belongs to Ω2 . Let us take any B any nbd of w0 in Ω2 and take inverse of B, f-1{B}. f-1{B} contains a nbd of z0 in Ω1.. In the case of Continuity the only difference is w0 =f(z0)),

45. Properties of Limits • If as z z0, lim f(z)  A, then A is unique If as z z0, lim f(z)  A and lim g(z)  B, then • lim [ f(z)  g(z) ] = A  B • lim f(z)g(z) = AB, and • lim f(z)/g(z) = A/B. if B  0.

46. Continuity • Definition. Let f(z) be a function such that f: C C. We call f(z) continuous at z0 iff: • F is defined in a neighborhood of z0, • The limit exists, and • A function f is said to be continuous on a set S if it is continuous at each point of S. If a function is not continuous at a point, then it is said to be singular at the point.

47. Test for Continuity of Functions it is true in a general metric space but not in general topological space. f: <S1,d1> <S2,d2> is continuous at s in S1. For all sn s f(sn) f(s)

48. Note on Continuity • One can show that f(z) approaches a limit precisely when its real and imaginary parts approach limits, and the continuity of f(z) is equivalent to the continuity of its real and imaginary parts.

49. Properties of Continuous Functions • If f(z) and g(z) are continuous at z0, then so are f(z)  g(z) and f(z)g(z). The quotient f(z)/g(z) is also continuous at z0provided that g(z0)  0. • Also, continuous functions map compact sets into compact sets.

50. Exercises • Find domain and range of the following functions and check their continuity: • f1(z)=z • f2(z)=|z| • f3(z)=z2 • f4(z)= • f5(z)=1/(z-2) • f6(z)=ez/log(z)/z1./2/cos(z)