
Recent Posts
Recent Comments
 Determinants II: Determinants of Order $n$  MathPhys Archive on Determinants I: Determinants of Order 2
 Inverses  MathPhys Archive on The Matrix Associated with a Linear Map
 The Matrix Associated with a Linear Map  MathPhys Archive on Linear Maps
 Introduction to Topology 3: Limit Points, Boundary Points, and Sequential Limits  MathPhys Archive on Introduction to Topology 1: Open and Closed Sets
 Parallel Transport, Holonomy, and Curvature  MathPhys Archive on Line Bundles
Archives
Categories
 Algebraic Topology
 Calculus
 Classical Differential Geometry
 College Algebra
 Differential Equations
 Differential Geometry
 Electromagnetism
 Engineering Mathematics
 Functions of a Complex Variable
 General Topology
 Homology
 Lie Groups and Lie Algebras
 Linear Algebra
 Mathematical Physics
 Partial Differential Equations
 Precalculus
 Quantum Mechanics
 Representation Theory
 Sage
 Trigonometry
 Uncategorized
Meta
Author Archives: lee
Residues at Poles
When $f(z)$ has a pole of order $m$, we may be able to find the residue of $f(z)$ at $z_0$ without expanding $f(z)$ into a Laurent series at $z=z_0$. This gives a great computational advantage. Suppose that $z_0$ is a … Continue reading
The Three Types of Isolated Singularities
Recall that if $f(z)$ has an isolated singularity at $z=z_0$, it may be represented by a Laurent series $$f(z)=\sum_{n=0}^\infty a_n(zz_0)^n+\frac{b_1}{zz_0}+\frac{b_2}{(zz_0)^2}+\cdots+\frac{b_n}{(zz_0)^n}+\cdots$$ in a puctured disk $0<zz_0<R$. The part of series that contains negative powers of $zz_0$ $$\frac{b_1}{zz_0}+\frac{b_2}{(zz_0)^2}+\cdots+\frac{b_n}{(zz_0)^n}+\cdots$$ is called the principal … Continue reading
More on Residues
Here and here, we studied how to evaluate the contour integral $\oint_C f(z)dz$ when $f(z)$ is analytic everywhere within and on the positively oriented simple closed contour $C$ except for a finite number of isolated singularities interior to $C$. The … Continue reading
Cauchy’s Residue Theorem
In here, we discussed that if a function $f(z)$ is analytic except at an isolated singularity $z_0$ interior to a positively oriented simple closed contour $C$, then $$\oint_C f(z)dz=2\pi i\mathrm{Res}_{z=z_0}f(z).$$ What if there are more than one isolated singularities of … Continue reading
Residues
Definition. A point $z_0$ is called a singular point or a singularity of a function $f$ if $f$ fails to be analytic at $z_0$ but is analytic at some point in every neighbourhood of $z_0$. A singularity is said to … Continue reading
Laurent Series
If a function fails to be analytic at a point $z_0$, we cannot apply Taylor’s theorem at that point. However, it may be possible to find a series representation for $f(z)$ involving both positive and negative powers of $zz_0$. Theorem … Continue reading
Taylor Series
Theorem. Suppose that a function $f$ is analytic throughout a disk $zz_0<R_0$ centered at $z_0$ and with radius $R_0$. Then $f(z)$ has the power series representation $$f(z)=\sum_{n=0}^\infty a_n(zz_0)^n\ (zz_0<R_0),$$ where $$a_n=\frac{f^{(n)}(z_0)}{n!}\ (n=0,1,2,\cdots).$$ Proof. First consider the case $z_0=0$ and show … Continue reading
Cauchy’s Inequality and Liouville’s Theorem
Suppose that a function $f$ is analytic inside and on a positively oriented circle $C_R$, centered at $z_0$ and with radius $R$. Then by Cauchy Integral Formula $$f^{(n)}(z_0)=\frac{n!}{2\pi i}\oint_{C_R}\frac{f(z)}{(zz_0)^{n+1}}dz.$$ Since $C_R$ is compact (i.e. closed and bounded) and $f(z)$ is … Continue reading
Morera’s Theorem
Cauchy’s Integral Theorem says that if a function $f(z)$ is analytic throughout some simply connected domain $D$, then for any contour $C$ in $D$, $\oint_C f(z)dz=0$. It turns out that the converse of Cauchy’s Theorem is also true, namely Theorem. … Continue reading
Cauchy Integral Formula
Suppose that $f(z)$ is analytic everywhere inside and on a simple closed contour $C$, taken in the positive sense. If $z_0$ is a point exterior to $C$, then by Cauchy’s Integral Theorem, $$\oint_C\frac{f(z)}{zz_0}dz=0.$$ Now, the question is what would be … Continue reading