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Tag Archives: complex analysis
Laurent Series with WolframAlpha
WolframAlpha can compute (simple) Laurent series: https://www.wolframalpha.com/input/?i=series+sin(z%5E-1) Series[Sin[z^(-1)], {z, 0, 5}] 1/z-1/(6 z^3)+1/(120 z^5)+O((1/z)^6) (Laurent series) (converges everywhere away from origin) Unfortunately, more “complex” (pun intended) Laurent series are not possible for WolframAlpha.
A holomorphic and injective function has nonzero derivative
This post proves that if is a function that is holomorphic (analytic) and injective, then for all in . The condition of having nonzero derivative is equivalent to the condition of conformal (preserves angles). Hence, this result can be stated … Continue reading
Rouche’s Theorem
Rouche’s Theorem If the complex-valued functions and are holomorphic inside and on some closed contour , with on , then and have the same number of zeroes inside , where each zero is counted as many times as its multiplicity. … Continue reading
Cauchy-Riemann Equations
Cauchy-Riemann Equations Let . The Cauchy-Riemann equations are: Alternative Form (Wirtinger Derivative) The Cauchy-Riemann equations can be written as a single equation where is the Wirtinger derivative with respect to the conjugate variable. Goursat’s Theorem Suppose is a complex-valued function … Continue reading
dz and dz bar: How to derive the Wirtinger derivatives
Something interesting in Complex Analysis is the Wirtinger derivatives: They are often simply defined as such, but one would be curious how to derive them, at least heuristically. How to derive Wirtinger derivatives It turns out we can derive them … Continue reading
Schwarz Lemma & Maximum Modulus Principle
Schwarz Lemma Let be the open unit disk in the complex plane centered at the origin and let be a holomorphic map such that . Then, for all and . Moreover, if for some non-zero or , then for some … Continue reading
Evaluation of Improper Integral via Complex Analysis
We are following the notation in Complex Variables and Applications (Brown and Churchill). The method of using complex analysis to evaluate integrals is to consider a very large semicircular region’s boundary, which consists of the segment of the real axis from … Continue reading
Evaluating Integrals using Complex Analysis
Our next few posts on complex analysis will focus on evaluating real integrals like using residue theory from Complex Analysis. This is something amazing about Complex Analysis, it can be used to solve integrals in real numbers, something which is … Continue reading
Cauchy’s Theorem
Cauchy’s Theorem: Let be the closed region consisting of all points interior to and on the simple closed contour . If is analytic in and is continuous in , (This is the precursor of Cauchy-Goursat Theorem, which allows us to … Continue reading
Mapping of Infinite Vertical Strip by the Exponential Function
This post is continued from a previous post on how to map an open region between two circles to a vertical strip. We wish to map the infinite vertical strip onto the upper half plane. Reference book is Complex Variables … Continue reading
Image of Vertical Strip under Inversion
This is a slight generalisation of Example 3 in Churchill’s Complex Variables and Applications. Consider the infinite vertical strip , under the transformation , where are of the same sign. When , note that by arguments similar to earlier analysis, … Continue reading
Linear Fractional Transformation (Mobius Transformation)
The transformation , with , and are complex constants, is called a linear fractional transformation, or Mobius transformation. One key property of linear fractional transformations is that it transforms circles and lines into circles and lines. Let us find the … Continue reading
