#set page(height: auto)
#let inv(n) = math.frac([1], n);
= Multiply
$
(a+b i)(c + d i)=\
a c + a d i + b c i + b d i^2=\
a c + (a d + b c)i - b d=\
a c - b d + (a d + b c)i
$
= Divide
$
frac(a + b i, c + d i)=\
frac((a + b i)(c - d i), (c + d i)(c + d i))=\
frac(a c + b d + (b c - a d)i, c^2 + d^2)=\
frac(a c + b d, c^2 + d^2) + frac(b c - a d, c^2 + d^2)i=\
$
= Square root
If $b$ is positive:
$
sqrt(z)=sqrt(frac(a+sqrt(a^2+b^2), 2))+sqrt(frac(-a+sqrt(a^2+b^2), 2))i
$
If $b$ is negative:
$
sqrt(z)=sqrt(frac(a+sqrt(a^2+b^2), 2))-sqrt(frac(-a+sqrt(a^2+b^2), 2))i
$
= Inverse
$
inv(z) =\
frac(a - b i, |z|^2)=\
frac(a, |z|^2) - frac(b, |z|^2) i
$
= Natural Logarithm
$
ln(z) =\
ln(|z|e^(i theta))=\
ln|z|+ln(e^(i theta))=\
ln|z|+i theta
$
= Exponentiation
$
z_1^(z_2)=\
z_1^(c + d i)=\
z_1^(c) z_1^(d i)=\
z_1^(c) e^(ln(z_1)d i)=\
$
= Trig
$
sin(z) & =sin(a)cosh(b) + cos(a)sinh(b)i \
cos(z) & =cos(a)cosh(b) - sin(a)sinh(b)i \
tan(z) & =frac(sin(z), cos(z)) \
cot(z) & =frac(cos(z), sin(z)) \
sec(z) & =inv(cos(z)) \
csc(z) & =inv(sin(z)) \
$
= Inverse Trig
$
arcsin(z) & =-i ln(sqrt(1-z^2) + i z) \
arccos(z) & =i ln(1/i sqrt(1-z^2) + z) \
arctan(z) & =arcsin(frac(z, sqrt(1+z^2))) \
"arccot"(z) & =arctan(inv(z)) \
"arcsec"(z) & =arccos(inv(z)) \
"arccsc"(z) & =arcsin(inv(z)) \
$
= Hyperbolic Trig
$
sinh(z) & =sinh(a)cos(b)+cosh(a)sin(b)i \
cosh(z) & =cosh(a)cos(b)+sinh(a)sin(b)i \
tanh(z) & =frac(sinh(z), cosh(z)) \
coth(z) & =frac(cosh(z), sinh(z)) \
sech(z) & =inv(cos(z)) \
csch(z) & =inv(sin(z)) \
$
= Inverse Hyperbolic Trig
$
"arcsinh"(z) & =ln(sqrt(z^2+1)+z) \
"arccosh"(z) & =ln(sqrt(z^2-1)+z) \
"arctanh"(z) & =1/2ln(frac(1+z, 1-z)) \
"arccoth"(z) & ="arctanh"(inv(z))=1/2ln(frac(1-z, 1+z)) \
"arcsech"(z) & ="arccosh"(inv(z)) \
"arccsch"(z) & ="arcsinh"(inv(z)) \
$