Singularities and Poles

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• Singularities of functions

  Suppose the functions f and g have isolated singularities at and these singularities are poles of m and n, respectively. What can be said about the singularities of the functions f+g, f.g and f/g?

• Zeroes of functions

  The poles are , where is an integer. For these pole, ( ) is an isolated singularity with order . is a removable singularity and the value of at this point is . (d) There is no zero. The poles are , each with order .

• Isolated singularities

  To find all the singularities of , we set , then we have . So the solutions are . We have the following observations. Therefore, has one removable singularity at and two poles at and .

• Complex Analysis and Singularities : If f : G -> C ( C here is complex plane) is analytic except for poles show that the poles of f cannot have limit point in G.

  58901 Complex Analysis and Singularities : If f : G -> C ( C here is complex plane) is analytic except for poles show that the poles of f cannot have limit point in G.

• Complex Function Calculus

  The function is: (1.45) The singularities occur at: (1.46) The roots are: (1.47) Only two singularities are inside the domain: (1.48) And the poles are simple We can write the function as: (1.49) Therefore the residue at

• Poles & Singularities

  46448 Poles & Singularities Classify the behavior at infinity (analytic, pole, zero, or essential singularity; if a zero or pole, give its order) of the following functions: f1(z) = (z^3 + i)/z f2(z) = e^(tan(1/z)) Let oo denote the infinity.

• Contour Integration & Cauchy Theorem

  at: (please see the attached file) (2.3) And the residues at these simple poles are: (please see the attached file) (2.4) (please see the attached file) (2.5) So there are four cases Case 1: (please see the attached file) The poles are

• Singular points and residues of 1/[(Cosh(z))-2exp(z)]

  of the function can be found, and we compute the residues at the poles.

• Complex Analysis / Singularities / Argument Principle : Let f be meromorphic on the region G and not constant; show that neither the poles nor the zeros of f have a limit point in G.

  59166 Complex Analysis / Singularities / Argument Principle Let f be meromorphic on the region G and not constant; show that neither the poles nor the zeros of f have a limit point in G.