17.4 Functions of a Complex Variable

Section 17.4: 10, 11, 18, 19, 21, 22, 23, 25, 26, 31, 34, 35, 37, 39.

A function $f$ that takes complex numbers as inputs and outputs, $w = f(z)$ where $z = x+iy$ and $w = u + iv$

• Function maps $z$ to $w$ by splitting into two real valued functions of two real variables ($x$ and $y$), the real and imaginary parts of $w$

• Since complex numbers are two dimensional, graphing these functions requires a four dimensional space

  

  Often represented by mappings or transformations from one plane to another.

Complex Functions as Flows

The complex number $f(z)$ is viewed as a vector field where each point $z$ in the domain has a vector $f(z)$ associated with it. This vector specifies the velocity of the fluid at that point

• Streamlines: if a particle is moving within this flow, its path can be represented parametrically by $x(t) + iy(t)$, where $x(t)$ and $y(t)$ are functions of time $t$
  • The solutions to the system of differential equations are called the streamlines of the flow
  • Curves that are tangent to the velocity vector of the flow at every point
• The particle’s path follows a system of differential equations $\frac{dx}{dt} = u(x,y)$ and $\frac{dy}{dt} = v(x,y)$
  • These equations describe how the position of the particle changes over time based on the velocity field determined by the complex function

Limits and Continuity

The limit is said to exist at a point $z_0$ if, for every $\epsilon$ greater than zero, there is a δ such that the value of the function is within ε of L whenever $z$ is within δ of $z_0$ but not equal to $z_0$.

Derivative

Derivative of a complex function $f$ at a point $z_0$ is defined similarly to that of a real function, but using complex number increments: $f’(z_0) = \lim {\triangle z → 0} \frac{f(z_0+ \triangle z) - f(z_0)}{\triangle z}$

Analytic Functions

Type of complex function that has derivatives of all orders in some region

• Differentiable at all points in some neighbourhood of that point

17.5 Cauchy-Riemann Equations

Section 17.5: 1, 3, 10, 11, 15, 25, 29, 30, 32.

If a complex function $f(z) = u(x,y) + iv(x,y)$ where $z = x+iy$ and $i$ is the imaginary unit, it is differentiable at a point $z$, then the first order partial derivatives of the real parts $u$ and imaginary parts $v$ exist and are:

$$ \begin{aligned}\frac{\partial u}{\partial x} &= \frac{\partial v}{\partial y} \\\frac{\partial u}{\partial y} &= -\frac{\partial v}{\partial x}\end{aligned} $$