bits and pieces

Author: jhobbs

content/analysis/complex-analysis/module-01-algebriac-properties.md

@@ -109,7 +109,9 @@ $$ z_1 \cdot z_2 = r_1 r_2 (cos(\theta_1 + \theta_2) + isin(\theta_1 + \theta_2)
 
 This can be interpreted geometrically that multiplying by a complex number causes rotation in the complex plane.
 
-We can use this proprety to easily compute powers of complex numbers, which is sometimes called **DeMoivre's theorem**:
+Note that when we take the conjugate of a number $z = r e^{i\theta}$ we get $\overline{z} = r^{-i\theta}.$ This is a number with the same magnitude as $z$ but with the opposite angle. On the unit circle, where all numbers have the same magnitude, we have $z \overline{z} = 1$ because multiplying $z$ by its conjugate rotates it exactly how much it was rotated, but in the opposite direction.
+
+We can use this property to easily compute powers of complex numbers, which is sometimes called **DeMoivre's theorem**:
 
 
 :::theorem "de Moivre's Theorem" {label: de-moivres-}
@@ -139,9 +141,9 @@ $$ X = x^2 - y^2, \quad Y = 2xy $$
 
 Now we have two equations and two unknowns, and when they are solved, we will have the two square roots of $Z$.
 
-By definition, the **principal square root** of $Z$ is the square root with the nonnegative real part.
+By definition, the **principal square root** of $Z$ is the square root with the non-negative real part.
 
-## Complex Quadratrics
+## Complex Quadratics
 
 We can use the quadratic formula to solve quadratics in a complex variable with complex coefficients, that is where
 

content/analysis/complex-analysis/module-08-elementary-functions.md

@@ -56,10 +56,10 @@ These functions are also $2 \pi$ periodic like their real counterparts.
 
 The real and imaginary parts of $\sin{z}$ and $\cos{z}$ can be expressed as
 
-$$ \sin{z} = \sin{(x + yi)} = \sin{x} \cosh{x} + \cos{x}\sinh{yi} $$
+$$ \sin{z} = \sin{(x + yi)} = \sin{x} \cosh{x} + i \cos{x}\sinh{y} $$
 
 
-$$ \cos{z} = \cos{(x + yi)} = \cos{x} \cosh{y} - \sin{x} \sinh{yi} $$
+$$ \cos{z} = \cos{(x + yi)} = \cos{x} \cosh{y} - i \sin{x} \sinh{y} $$
 
 $\cos{z}$ and $\sin{z}$ are entire functions. They have the same zeros as the real versions and the expected derivatives.
 
@@ -81,9 +81,9 @@ $$ \sinh{(iz)} = i\sin{z}, \quad \cosh{iz} = \cos{z}. $$
 
 The real and imaginary parts of $\sinh{z}$ and $\cosh{z}$ are given as
 
-$$ \sinh{z} = \cos{y} \sinh{x} + \sin{y} \cosh{xi}, $$
+$$ \sinh{z} = \cos{y} \sinh{x} + i \sin{y} \cosh{x}, $$
 
-$$ \cosh{z} = \cos{y} \cosh{x} + \sin{y} \sinh{xi}. $$
+$$ \cosh{z} = \cos{y} \cosh{x} + i \sin{y} \sinh{x}. $$
 
 The remaining complex hyperbolic functions are defined as usual
 
@@ -111,11 +111,11 @@ The principal logarithm is defined using the principal argument, such that
 
 $$ \Log{z} = \ln{|z|} + (\Arg{z})i $$
 
-and is analytic in the domain $\|z\| > 0, -\pi < \Arg{z} < \pi.$ It is also called the prinicpal branch of $\log{z}$.
+and is analytic in the domain $\|z\| > 0, -\pi < \Arg{z} < \pi.$ It is also called the principal branch of $\log{z}$.
 
 Points on the negative real aixs and $z = 0$ are singularities of $\Log{z}$, but they are not isolated singularities.
 
-The deriviative of $\Log{z}$ is
+The derivative of $\Log{z}$ is
 
 $$ \frac{d}{dz} \Log{z} = \frac{1}{z} $$
 
@@ -125,4 +125,10 @@ $$ \log_\phi{z} = \ln{|z|} + (\arg_\phi{z})i. $$
 
 These branches are analytic on any domain that does not contain $z = 0$ (the branch point) or points on the branch cut (the half-line through $z=0$ making an angle of $\phi$ radians ith the positive real axis.)
 
-We can define branch cuts that aren't straight lines too. For example, branch points of the function $\Log{\left \[ f(z)\right \]}$ are the zeroes of $f(z)$ and branch cuts are where $f(z)$ is real and negative.
+We can define branch cuts that aren't straight lines too. For example, branch points of the function $\Log{ \left [ f(z) \right ] }$ are the zeroes of $f(z)$ and branch cuts are where $f(z)$ is real and negative.
+
+## General Powers of z
+
+For a complex number $a,$
+
+$$ a^z = e^{z * \ln{a}}. $$

content/analysis/complex-analysis/module-20-taylor-and-maclaurin-series.md

@@ -96,7 +96,9 @@ Here we can see that $w$ contributes only a factor of $|w|,$ which doesn't depen
 
 $$ L = \lim_{n \to \infty} \left | \frac{a_{n+1}}{a_n} \right | \implies \text{series converges if} |w| < 1/L. $$
 
-So, our radius of convergence here is $1/L.$ Note that if we had something like $w = (z - z_0)^{mn},$ then our radius of convergence would be $(1/L)^{1/m}.$
+So, our radius of convergence for our $w$ series here is $1/L.$ We have to bring $z$ back, so for example, if we had something like $w = (z - z_0)^{mn},$ then our radius of convergence would be $(1/L)^{1/m}.$ We get that by
+
+$$ |w| < R \implies |z^m| < R \implies |z| < R^{1/m}. $$
 
 Similarly with the root test, we end up with something like
 

content/analysis/complex-analysis/module-24-real-integrals.md

@@ -86,7 +86,7 @@ $$ I = \frac{1}{2} * 2 \pi i * (\Res{z \to z_1}f(z) + \Res{z \to z_2}f(z)) = \fr
 
 Putting this all together, we have, in general, with the conditions specified,
 
-$$ \int_{-\infty}^{\infty} f(x) dx = 2 * \pi * \sum \Res f(z). $$
+$$ \int_{-\infty}^{\infty} f(x) dx = 2 \pi i \sum \Res f(z). $$
 
 ## Improper Integrals with Simple Poles on the Real Axis