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General Understanding of Terms [ Complex Analysis ]

  1. Jun 29, 2015 #1

    RJLiberator

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    Gold Member

    Hi all,

    I was unsure where to put this thread as I read the main topic title in the topology/analysis forums and decided to post it here.

    I am looking for a chart/graph/website that helps me understand the basic terms such as:

    -neighborhoods
    -Boundary points
    -Singularity points
    - "Function is analytic"
    - "Function is entire
    - Function is harmonic

    and anything similar.

    My ultimate goal is to create some posters to hang up in my room based on the information to help me learn the terms. So for this, I need some well organized, to the point, material to work off of since I don't understand the terms currently.

    At best, I am hoping to find a visual understanding of the terms. If you have any good online resources please post them here. It's been difficult for me to find some via Google without really understanding the terms.

    Thanks.
     
  2. jcsd
  3. Jun 30, 2015 #2
    Anything specific you have questions about?
     
  4. Jun 30, 2015 #3

    RJLiberator

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    Gold Member

    Yes.

    We went over terms and geographic understandings of sorts of:

    Boundary Points
    "Neighborhoods" of functions
    Singular points
    Any term in that realm. I was hoping to find a summary of these terms/definitions with graphs/charts.

    We also went over when a function is entire, analytic, and/or harmonic. I would love something that is short and helps me understand these definitions as well. It seems the material I am finding on these terms is very long and I am unable to simplify it into a poster or simple understanding of sorts.
     
  5. Jun 30, 2015 #4

    Svein

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    No wonder. It is not simple and there is no shortcut to understanding. I could very well create a short definition list for you, but that list would have to refer to other terms that you would not understand. Examples:

    • In a metric space, x is an interior point of a set A if x∈A and there exists a δ>0 such that all points y with d(x, y)<δ belongs to A.
    • A point z is a boundary point of a set A if z∈A and for all δ>0 there exist points y∉A with d(z, y)<δ
    • A complex function f(z) where z = x + iy (and f(z) = u(z)+iv(z)) is analytic if [itex]\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} [/itex] and [itex]\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x} [/itex]
    Please read up the rest in Ahlfors.
     
  6. Jun 30, 2015 #5

    RJLiberator

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    Gold Member

    I understand.

    But perhaps there is no sets of images available?

    Let me give an example.

    http://www.solitaryroad.com/c612.html

    Images like this on the right describing the points is ideally what I am looking for.
     
  7. Jun 30, 2015 #6

    mathwonk

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    here isa brief extract from some class notes i taught:

    Holomorphic versus analytic functions:

    1. A function f is called "holomorphic" in an open set U if f is complex differentiable everywhere in U.

    2. f is called C^1 - holomorphic in U, if it is holomorphic and also the complex derivative is continuous everywhere in U.

    3. f is called C^infinity - holomorphic in U, if f has infinitely many continuous complex derivatives at every point of U.

    4. f is called analytic in U, if near every point of U f is represented by a power series.

    The main theorem is that all these are equivalent. In particular all holomorphic functions in U are actually analytic in U.
     
  8. Jun 30, 2015 #7

    mathwonk

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    here is a review of complex differentiability:

    I. Complex differentiability and the Cauchy Riemann equations

    We are studying the behavior of a special class of differentiable functions f:C-->C, where C is the complex number plane. The complex number plane C ≈ R^2 is just the real number plane equipped with an additional multiplication, and our complex differentiable functions are a subclass of the real differentiable functions from R^2 to R^2, familiar from vector calculus. When thought of as a complex number, the point (a,b) of R^2 is written as a+bi.

    The usual definition of complex differentiability is this: f is holomorphic at p, if it is defined on some neighborhood of p, and the limit of the difference quotient

    [f(z)-f(p)]/(z-p) exists as z approaches p. If f'(p) is that limit, then the linear function L(h) = f'(p).h, is the best complex linear approximation

    to the function f(p+h) - f(p), both considered as functions of h.


    Here is the relation to real differentiability.

    Recall that a function f:R^2-->R^2 is differentiable at the point p = (a,b) in the real sense, if it has a good approximation near (a,b) by a real linear transformation L. [“Good approximation” means the graph of L(h) is tangent at (0,0) to the graph of f(p+h) - f(p).] A function f:C-->C is said to be complex differentiable, or holomorphic, at the point a+bi, if it is differentiable in the real sense, and if moreover the linear function L that approximates it near a+bi is not just real - linear but complex linear. That is, not only is L(v+w) = L(v)+L(w) for all v,w in C, and L(av) = aL(v) for all v in C and all a in R, but also L(cv) = cL(v) for all v in C and all c in C.


    We know all real linear maps of R^2 are represented by real 2by2 matrices, so all complex linear maps also have such matrices, but because of the extra requirement for complex linearity, the real matrices representing complex linear maps have a special form. A real linear map L is also complex linear if L(iv) = iL(v) for all v in C, so we ask next what this says about the real matrix for L.


    The real matrix for L has the coordinates of L(<1,0>) in its first column, and the coordinates of L(<0,1>) in its second column. Now multiplication by i is itself a real linear map, taking 1 = <1,0> to i = <0,1>, and taking <0,1> = i, to -1 =

    <-1,0>, so the matrix representing multiplication by i has rows [ 0 -1], [ 1 0]. For a real 2x2 matrix M to represent a complex linear map, it must commute with this matrix, which forces the matrix M to have rows of form [ a -b], [ b a]. Here a+bi is the usual complex derivative, as a complex number. Since the entries of this matrix are just the partials of the real component functions u,v of f = u+iv, this says those partials satisfy the Cauchy Riemann equations, ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x.
     
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