#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\
$
= Inverse Trig
$
arcsin(z)&=-i ln(sqrt(1-z^2) + i z)\
arccos(z)&=i ln(-i sqrt(1-z^2) + z)\
arctan(z)&=inv(2i)ln(frac(1+i z,1-i z))\
"arccot"(z)&=arctan(inv(z)) = inv(2i)ln(frac(z+i,z-i))\
"arcsec"(z)&=arccos(inv(z)) = -i ln(sqrt(1/z^2-1) + 1/z)\
"arccsc"(z)&=arcsin(inv(z))=-i ln(sqrt(1-1/z^2) + i/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\
$
= 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(z+1,z-1))\
"arcsech"(z)&="arccosh"(inv(z))=ln(sqrt(1/z^2-1)+1/z)\
"arccsch"(z)&="arcsinh"(inv(z))=ln(sqrt(1/z^2+1)+1/z)\
$