Page Description Equation
99 polynomial function \[ p_n\left(z\right)=a_0+a_1z+a_2z^2+\cdots+a_nz^n \]
99 rational function \[ R_{m,n}\left(z\right) =\frac{a_0+a_1z+a_2z^2+\cdots+a_mz^m} {b_0+b_1z+b_2z^2+\cdots+b_nz^n} \]
103 Taylor form of a polynomial centered at $z_0.$ Called the Maclaurin form when $z_0=0.$ \[ p_n\left(z\right) =\sum_{k=0}^{n}{\frac{p_n^{\left(k\right)}\left(z_0\right)} {k!}\left(z-z_0\right)^k} \]
111 exponential function \[ e^z=e^x\left(\cos{y}+i\sin{y}\right) \]
111 derivative of the exponential function \[ \frac{d}{dz}e^z=e^z \]
111 modulus of the exponential function \[ \abs{e^z}=e^x \]
111 argument of the exponential function \[ \arg{e^z}=y+2k\pi k=0,\pm1,\pm2,\ldots \]
111 theorem \[ e^z=1\Leftrightarrow z=2k\pi i \]
111 period of the complex exponential function. periodic with period $2\pi i.$ \[ e^{z_1}=e^{z_2}\Leftrightarrow z_1=z_2+2k\pi i \]
113 definition of $\cos z$ \[ \cos{z}=\frac{e^{iz}+e^{-iz}}{2} \]
113 definition of $\sin z$ \[ \sin{z}=\frac{e^{iz}-e^{-iz}}{2i} \]
113 derivative of $\sin z.$ \[ \frac{d}{dz}\sin{z}=\cos{z} \]
113 derivative of $\cos z.$ \[ \frac{d}{dz}\cos{z}=-\sin{z} \]
114 the only zeros of $\cos$ are its real zeros. \[ \cos{z}=0\Leftrightarrow z=\frac{\pi}{2}+k\pi \]
114 the only zeros of $\sin$ are its real zeros. \[ \sin{z}=0\Leftrightarrow z=k\pi \]
114 definition of $\tan z$ \[ \tan{z}=\frac{\sin{z}}{\cos{z}} \]
114 definition of $\cot z$ \[ \cot{z}=\frac{\cos{z}}{\sin{z}} \]
114 definition of $\sec z$ \[ \sec{z}=\frac{1}{\cos{z}} \]
114 definition of $\csc z$ \[ \csc{z}=\frac{1}{\sin{z}} \]
114 derivative of $\tan z$ \[ \frac{d}{dz}\tan{z}={\sec}^2{z} \]
114 derivative of $\cot z$ \[ \frac{d}{dz}\cot{z}=-\csc^2{z} \]
114 derivative of $\csc z$ \[ \frac{d}{dz}\csc{z}=-\csc{z}\cot{z} \]
114 derivative of $\sinh z$ \[ \sinh{z}=\frac{e^z-e^{-z}}{2} \]
derivative of $\cosh z$ \[ \cosh{z}=\frac{e^z+e^{-z}}{2} \]
131 $n$ an integer \[ z^n=\left(e^{\log{z}}\right)^n=e^{n\log{z}} \]
132 $\alpha=$ complex constant, $z\ne0$ \[ z^\alpha=\left(e^{\log{z}}\right)^\alpha =e^{\alpha\log{z}}\left(z\neq0\right) \]

Notes

Page Notes
99 The identically zero polynomial is assigned a degree of $–\infty.$
100 deflated polynomial
101 Fundamental theorem of algebra. Every nonconstant polynomial with complex coefficients has at least one zero in $\C.$
104 If $p_n\left(z\right)$ has a zero of multiplicity $k$ at $z_0,$ then $p_n^{\left(k\right)}\left(z_0\right)\neq0,$ while $p_n^{\left(j\right)}\left(z_0\right)=0$ for $0\le j\lt k$
104 The poles of a rational function are the zeros of its new denominator when all common factors have been cancelled.
104 A rational function maps neighborhoods of its poles into neighborhoods of infinity, in the sense of section 1.7.
105 Partial fraction decomposition.
111 Unlike in the real case, the complex exponential function is not one-to-one.
112 fundamental region for $e^z$
Notice the difference between \[ \cos{\theta} =Re{e^{i\theta}} =\frac{e^{i\theta}+e^{-i\theta}}{2}\\ Re{z}=z+z2=zeiθ+e-iθ2=z\cosθ\\ \cos{z}=\frac{e^{iz}+e^{-iz}}{2}\\ \cosh{z}=\frac{e^z+e^{-z}}{2}\\ \cosh{x}=\frac{e^x+e^{-x}}{2},\,x\text{ real} \] Similar differences exist for $\Im z, \sin z, \sinh z, \sinh x.$
121 $\Log\abs{z}$ is harmonic on $R^2\backslash\left\{0\right\}$
121 $\Arg z$ is harmonic on $R^2\backslash\left(-\infty,0\right]$
122 formal definition of branch.
113 the usual trig identities hold for complex trig