Complex Variable Class - Undergraduate 500 Level

Use the formal method, involving an infinite series of residues and illustrated in the examples. 7. F(s) = 1/(s cosh (s^½)) Please see attachment for full question.

Punctured Disk : Poles, Taylor Series, Laurent Series and Power Series

Suppose that a function F(s) has a pole of order m at s=s0, with a Laurent series expansion ... in the punctured disk 0<|s - s0| ...continues

It is an explanation of the Cauchy-Hadamard theorem on power series. Cauchy- Hadamard Theorem :- ∞ For every power series ∑ anzn there exist a number R, 0 ≤ R < ∞ n=0 called the radius of convergence with the following properties: (i) The series converges absolutely for every |z| < R (ii) If 0 ≤ ρ < R, the series converges uniformly for |z| ≤ ρ (iii) If |z| > R, the terms of the series are unbounded and the series is consequently divergent.

Complex Variable Cauchy- Hadamard Theorem :- ...continues

It is an explanation of a theorem in radius of convergence of power series. ∞ ∞ Theorem:- ∑ anzn is a power series and ∑ nanzn – 1 is the power series obtained n=0 n=0 by differentiating the first series term by term. Then the derived series has the same radius of convergence as the original series.

Complex Variable Radius of Convergence of the Power Series ...continues

It is an explanation of a theorem in radius of convergence of power series. ∞ ∞ Theorem:- ∑ anzn is a power series and ∑ nanzn – 1 is the power series obtained n=0 n=0 by differentiating the first series term by term. Then the derived series has the same radius of convergence as the original series.

Complex Variable Radius of Convergence of the Power Series ...continues

Linear Transformations : Polar and Rectangular Coordinates

Find the region onto which the half plane y > 0 is mapped by the transformation w = (1+i)z by using polar and rectangular coordinates. Please see the attachment for complete question.

Semi-Infinite Strip : Image

Find the image of the semi-infinite strip x > 0, 0 < y < 2 when w = iz + 1. Sketch the strip and its image . {Please see attachment}

Transformation of Half Plane to Interior of a Circle

2. Show that when c1<0 the image of the half plane x

Infinite Strip under Given Transformation

Find the image of the infinite strip 0 < y < 1/(2c) under the transformation w = 1/z. Sketch the strip and its image. {Please see attachment.}

Hyperbola; Lemniscate

10. By writing {see attachment}, show that the mapping w = 1/z transforms the hyperbola {see attachment} into the lemniscate {see attachment}