Page Description Equation
54 complex function of a complex variable \[ w=u\left(x,y\right)+iv\left(x,y\right) \]
55 inversion mapping \[ f\left(z\right)=\frac{1}{z} \]
58 Limit of a sequence of complex numbers. Let $\{z_n\}_1^\infty$ be a sequence of complex numbers and $\varepsilon\in\R,\,$ $n,N\in\Z.$ If \[ \forall\varepsilon\gt0,\exists N,\forall n\gt N\\ \abs{z_n-z_0}\lt\varepsilon \] then the sequence is said to have the limit $z_0$ or to converge to $z_0$ and we write this as shown. \[ \lim\limits_{n\rightarrow\infty}z_n=z_0\\ \text{or}\\ z_n\rightarrow z_0\text{ as } n\rightarrow\infty \]
59 limit of a complex function of a complex number. Let $z,z_0,\omega_0,f\in\C$, $f$ a complex-valued function defined in some neighborhood of $z_0,$ with the possible exception of the point $z_0$ itself, and $\varepsilon,\delta\in\R.$ If \[ (\forall\varepsilon\gt0) (\exists\delta\gt0) (\forall z)\\ (\abs{z-z_0}\lt\delta\Rightarrow \abs{f(z)-\omega_0}\lt\varepsilon) \] we say that the limit of $f(z)$ as $z$ approaches $z_0$ is the number $\omega_0$ and write this as shown. \[ \lim\limits_{z\rightarrow z_0}f\left(z\right)=\omega_0\\ \text{or}\\ f\left(z\right)\rightarrow \omega_0\text{ as }z\rightarrow z_0 \]
61 continuity of a complex function Let $f$ be a function defined in a neighborhood of $z_0.$ Then $f$ is continuous at $z_0$ if $\ldots$ \[ \lim\limits_{z\rightarrow z_0}f\left(z\right) =f\left(z_0\right) \]
61 limit rules where $\lim\limits_{z\rightarrow z_0} f(z)=A$ and $\lim\limits_{z\rightarrow z_0} g(z)=B.$ The rules here state that the limit of the sum, product or quotient of two functions is the sum, product, or quotient of their limits, respectively. \[ \begin{array}{l} \textbf{Sum Rule}\\ \lim\limits_{z\rightarrow z_0}\left(f\left(z\right)\pm g\left(z\right)\right)=A\pm B\\ \textbf{Product Rule}\\ \lim\limits_{z\rightarrow z_0}f\left(z\right)g\left(z\right)=AB\\ \textbf{Quotient Rule}\\ \lim\limits_{z\rightarrow z_0}\frac{f\left(z\right)}{g\left(z\right)} =\frac{A}{B},\ B\neq0 \end{array} \]
61 continuity rules where $\lim\limits_{z\rightarrow z_0} f(z)=f(z_0)=A$ and $\lim\limits_{z\rightarrow z_0} g(z)=g(z_0)=B.$ That is, $f$ and $g$ are continuous. The rules here state that if $f$ and $g$ are continuous, then their sums, products and quotients are. \[ \begin{array}{l} \textbf{Sum Rule}\\ \lim\limits_{z\rightarrow z_0}\left(f\left(z\right)\pm g\left(z\right)\right)=A\pm B\\ \textbf{Product Rule}\\ \lim\limits_{z\rightarrow z_0}f\left(z\right)g\left(z\right)=AB\\ \textbf{Quotient Rule}\\ \lim\limits_{z\rightarrow z_0}\frac{f\left(z\right)}{g\left(z\right)} =\frac{A}{B},\ B\neq0 \end{array} \]
61 Polynomial functions in $z$ are continuous on the whole plane. \[ P(z)=a_0+a_1z+a_2z^2+\cdots+a_n z^n\\ \Rightarrow \lim\limits_{z\rightarrow z_0}P(z)=P(z_0) \]
62 Rational functions in $z$ are continuous at each point in the plane where the denominator is not zero. \[ f(z)=\frac{P(z)}{D(z)} =\frac{a_0+a_1z+a_2z^2+\cdots+a_n z^n} {b_0+b_1z+b_2z^2+\cdots+b_n z^n}\\ \text{and }D(z_0)\ne0\Rightarrow \lim\limits_{z\rightarrow z_0}f(z)=f(z_0), \]
62 infinite limit of a sequence of complex numbers. Let $\{z_n\}_1^\infty$ be a sequence of complex numbers and $M\in\R,\,$ $n,N\in\Z.$ If \[ (\forall M\gt0)(\exists N)(\forall n\gt N)\\ (\abs{z_n}\gt M) \] then the sequence is said to have an infinite limit and we write this as shown. \[ \lim\limits_{n\rightarrow\infty}z_n=\infty\\ \text{or}\\ z_n\rightarrow\infty\text{ as } n\rightarrow\infty \]
62 infinite limit of a complex function of a complex number. Let $z,z_0,f\in\C$, $f$ a complex-valued function defined in some neighborhood of $z_0,$ with the possible exception of the point $z_0$ itself, and $M,\delta\in\R.$ If \[ (\forall M\gt0) (\exists\delta\gt0) (\forall z)\\ (\abs{z-z_0}\lt\delta\Rightarrow \abs{f(z)}\gt M) \] we say that the limit of $f(z)$ as $z$ approaches $z_0$ is $\infty$ and write this as shown. \[ \lim\limits_{z\rightarrow z_0}f\left(z\right)=\infty\\ \text{or}\\ f\left(z\right)\rightarrow \infty\text{ as }z\rightarrow z_0 \]
67 Let $f$ be defined in a neighborhood of $z_0.$ Then the derivative of $f$ at $z_0$ is given, provided the limit exists. $f$ is said to be differentiable at $z_0.$ \[ \frac{df}{dz}\equiv f'(z_0) :=\lim\limits_{\Delta z\rightarrow0} \frac{f(z_0+\Delta z)-f(z_0)} {\Delta z} \]
68
69		 differentiation rules \[ \begin{array}{l} \textbf{Power Rule}\\ \frac{d}{dz}z^n=nz^{n-1}\\ \textbf{Sum Rule}\\ \left(f\pm g\right)'\left(z\right) =f'\left(z\right)\pm g'\left(z\right)\\ \textbf{Constant Product Rule}\\ \left(cf\right)'\left(z\right)=cf'\left(z\right)\\ \textbf{Product Rule}\\ \left(fg\right)'\left(z\right) =f'\left(z\right)g\left(z\right)+f\left(z\right)g'\left(z\right)\\ \textbf{Quotient Rule}\\ \left(\frac{f}{g}\right)'\left(z\right) =\frac{g\left(z\right)f'\left(z\right)-f\left(z\right)g'\left(z\right)} {g\left(z\right)^2},\ g\left(z\right)\neq0\\ \end{array} \]
69 chain rule \[ \frac{d}{dz}f\left(g\left(z\right)\right)=f'\left(g\left(z\right)\right)g'\left(z\right) \]
73 Cauchy-Riemann equations \[ \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x} \]
73 derivative \[ f'\left(z_0\right)=\frac{\partial u}{\partial x}\left(x_0,y_0\right)+i\frac{\partial v}{\partial x}\left(x_0,y_0\right) \]
78 Jacobian from the $xy$-plane to the $uv$-plane where $u=u(x,y)$ and $v=v(x,y)$ and the partial derivatives are all evaluated at $(x_0,y_0).$ \[ J(x_0,y_0)= \begin{vmatrix} \frac{\partial u}{\partial x} &\frac{\partial u}{\partial y}\\ \frac{\partial v}{\partial x} &\frac{\partial v}{\partial y}\\ \end{vmatrix} \]
79 Laplace equation in two dimensions \[ \nabla^2\varphi=\frac{\partial^2\varphi}{\partial x^2}+\frac{\partial^2\varphi}{\partial y^2}=0 \]

Notes

Page Notes
54 Graphical representation of a complex function.
53 Definitions of function, domain of definition, mapping, image, range.
57
58
  • Joukowski mapping
  • translation mapping
  • rotation mapping
  • magnification mapping
  • reduction mapping
  • composite mapping
  • linear transformations
  • transformations involving the Riemann Sphere and
  • stereographic projection.
61 polynomial and rational functions are continuous.
62 removable discontinuity
62 point at infinity
69 Differentiability implies continuity.
70 A complex-valued function $f(z)$ is said to be analytic on an open set $G$ if it has a derivative at every point of $G.$
70 $f(z)$ is analytic at a point $z_0$ if $f(z)$ is analytic in some neighborhood of $z_0.$
70 A point where $f$ is not analytic but which is the limit of points where $f$ is analytic is known as a singular point or singularity.
70 If $f(z)$ is analytic on the whole complex plane, then it is said to be entire.
70 All analytic functions can be written in terms of $z$ alone (no $x,y,$ or $z$).
69 For purposes of differentiation, polynomial and rational functions in $z$ can be treated as if $z$ were a real variable.
73 If $f$ is analytic in an open set $G,$ then the Cauchy-Riemann equations must hold at every point of $G.$
74 If the first partial derivatives of $f$ are continuous and satisfy the Cauchy-Riemann equations at all points of $G,$ then $f$ is analytic in $G.$
77 An analytic function $f$ must be constant when any one of the following conditions hold in a domain $D.$
76 If $f(z)$ is analytic in a domain $D$ and if $f'\left(z\right)=0$ everywhere in $D,$ then $f(z)$ is constant in $D.$
79 A real valued function $\varphi\left(x,y\right)$ is said to be harmonic in a domain $D$ if all its second-order partial derivatives are continuous in $D$ and if, at each point of $D,$ $\phi$ satisfies the Laplace equation.
79 If $f\left(z\right)=u\left(x,y\right)+iv\left(x,y\right)$ is analytic in a domain $D,$ then each of the functions $u\left(x,y\right)$ and $v\left(x,y\right)$ is harmonic in $D$ and $v$ is called the harmonic conjugate of $u.$
79 If $u\left(x,y\right)$ is a harmonic function in an open disk $D,$ then there is a function $v\left(x,y\right)$ so that $u\left(x,y\right)+iv\left(x,y\right)$ is analytic in $D.$
81 Finding a harmonic conjugate in an arbitrary domain may not always be possible. See problem 21 for an example, when the domain is a punctured disk.
79 isotimic curves; level curves
81 electrostatic potential; equipotentials; isotherms
83 The level curves of the real and imaginary parts of an analytic function $f$ will always intersect at right angles, unless $f'\left(z\right)=0$ at the point of intersection. Thus, level curves of harmonic functions and their harmonic conjugates intersect at right angles.