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246A: Complex Analysis, Notes 2 – Meromorphic Functions and Properties of Analytic Functions

In Mathematics on February 10, 2011 at 2:24 am

We continue with a discussion about meromorphic functions and the properties of analytic functions. Later notes will consider the Riemann mapping theorem, harmonic functions and the Dirichlet problem among other topics.

Definition 2.1 A function {f} on an open set {\Omega} is meromorphic if there exists a discrete set of points {S = \left\{z: z \in \Omega\right\}} such that {f} is holomorphic on {\Omega\setminus S} and has poles at each {z \in S}. Furthermore, {f} is meromorphic in the extended complex plane if {F(z) = f(1/z)} is either meromorphic or holomorphic at {0}. In this case we say that {f} has a pole or is holomorphic at infinity.

By collecting results from the previous section, we are immediately led to the following proposition regarding the Laurent expansions of complex valued functions.

Proposition 2.2 Let {S} be the discrete set of singularities of a complex function {f:\Omega \rightarrow \mathbb{C}} where {\Omega} is an open set in {\mathbb{C}}. For a fixed {z_{0} \in S}, suppose the Laurent expansion for {f} in an annulus about {z_{0}} is given by {\sum_{-\infty}^{\infty}a_{n}(z-z_{0})^{n}}. Then,

  1. The function {f} has a removable singularity at {z_{0}} if and only if {a_{n} = 0} for all {n < 0}.
  2. The function {f} has a pole at {z_{0}} if and only if there exists {N \in \mathbb{Z}} with {N < 0} such that {a_{n} = 0} for all {n < N}; that is, the Laurent expansion of {f} about {z_{0}} has only finitely many negative terms.
  3. The function {f} has an essential singularity at {z_{0}} if and only if the Laurent expansion of {f} about {z_{0}} has infinitely many negative terms.
  4. Furthermore, {f} is meromorphic on the extended complex plane if and only if there exists {N \in \mathbb{Z}^{+}} such that {a_{n} = 0} for {n > N}.


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246A: Complex Analysis, Notes 1 – Analytic functions, Cauchy’s formula and singularities.

In Mathematics on January 4, 2011 at 7:36 pm

The following series of posts comprises our introduction to complex analysis as taught by Professor Rowan Killip at the University of California, Los Angeles, during the Fall quarter of 2009. Where necessary, course notes have been supplemented with details written by the authors of this website using assistance from Complex Analysis by Elias Stein and Rami Shakarchi. The basic properties of complex numbers will be assumed allowing us to begin with the definition of a holomorphic (or complex-differentiable) function, the central notion in our study of complex analysis.

The basic properties of complex numbers will be assumed, allowing us to begin with the definition of a holomorphic (or complex-differentiable) function, the central notion in our study of complex analysis.

Definition 1.1 Suppose {\Omega \in \mathbb{C}} is an open set and {f:\Omega \rightarrow \mathbb{C}}. We say {f} is holomorphic (or complex-differentiable) at {z_0 \in \Omega} if there exists {f^{\prime}(z_0) \in \mathbb{C}: f(z) = f(z_0) + f^{\prime}(z_0)(z - z_{0}) + o(\left |z-z_0\right |).} We say {f} is holomorphic on {\Omega} if it has this property for all {z \in \Omega}.

We can rewrite this formula in terms of the real and imaginary parts of {f} to surmise the relationship between complex differentiability and real analytic differentiability. Let {z, z_{0} \in \mathbb{C}} with {z = x+iy} and {z_{0} = x_0 + iy_{0}}, {x, y, x_0, y_0 \in \mathbb{R}} and write {f(z) = u(x,y) + iv(x,y)} where {u,v: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}}. Then,

\displaystyle  \left[ \begin{array}{cc} u(x,y) \\ v(x,y) \end{array} \right] = \left[ \begin{array}{cc} u(x_0, y_0) \\ v(x_0, y_0) \end{array} \right] + \left[ \begin{array}{cc} {\rm Re}f^{\prime}(z_{0}) & -{\rm Im}f^{\prime}(z_{0}) \\ {\rm Im}f^{\prime}(z_{0}) & {\rm Re}f^{\prime}(z_{0}) \end{array} \right] \left[ \begin{array}{cc} x-x_{0} \\ y-y_{0} \end{array} \right] + o(\left |x-x_{0}\right | + \left |y-y_{0}\right |).

We first notice that this is stronger than the differentiability of the real map {(x, y) \mapsto (u(x,y), v(x,y))} in {\mathbb{R}^2 \rightarrow \mathbb{R}^2}. In the real, multivariable case, the derivative of this map is a linear operator, namely, the Jacobian, {J_{f}(x,y)}; in our equation above, the {2 \times 2} matrix on the right hand side is {J_{f}(x,y)}. Clearly, it is endowed with a distinct structure summarized in the following proposition.

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