This can be usefull for an implementation of complex arithmetics using base formulas to evalute base functions.

COMPLEX ARITHMETIC ML-04

algorithms :

X = a + bi

Y = c + di

X + Y = (a + c) + i x (b + d)

X - Y = (a - c) + i x (b - d)

X x Y = (ac - bd) + i x (ad + bc)

X / Y = [(ac + bd) / (cc + dd)] + i x [(bc - ad) / (cc + dd)] , Y != 0

Y ^ X = e^(X x ln(Y)) , Y != 0

log[Y]X = ln(X) / ln(Y) , X != 0, Y != 0 , ln(Y) != 0

Y ^ (1/X) = e^(ln(Y) / X) , X != 0 , Y != 0

COMPLEX FUNCTIONS ML-05

algorithms :

X = a + bi

Polar Form of X :

r = sqrt( (a x a) + (b x b) )

@ in (-PI/2 <= @ <= 3xPI/2)

@ = atan2( b / a ) , a != 0

@ = PI / 2 , a == 0 , b > 0

@ = - PI / 2 , a == 0 , b < 0

X ^ 2 = r^2 x ( cos(2@) + i x sin(2@) )

X ^ 1/2 = r^1/2 x ( cos(@/2) + i x sin(@/2) )

1 / X = 1 / ( a + bi ) = a / r^2 - i x b / r^2 , X != 0

e ^ X = e^a x cos(b) + i x e^a x sin(b)

ln(X) = ln(r) + i@ , X != 0

COMPLEX TRIG FUNCTIONS ML-06

algorithms :

X = a + bi

sin(X) = (e^iX - 1/e^iX) / 2i , e^iX != 0

cos(X) = (e^iX + 1/e^iX) / 2 , e^iX != 0

tan(X) = sin(X) / cos(X) , cos(X) != 0

asin(X) = ln( iX + sqrt(1 - X^2) ) / i

acos(X) = ln( X + i x sqrt(1 - X^2) ) / i

atan(X) = ln( (1 + iX) / (1 - iX) ) / 2i , (1+iX) != 0 , (1-iX) != 0

Reference: Texas Instruments TI58-59 , Master Library (TI58-59-MasterLibraryManual.pdf) , page 18-23

algorithms :

X = a + bi

sinh(X) = (e^X - 1/e^X) / 2 , e^X != 0

cosh(X) = (e^X + 1/e^X) / 2 , e^X != 0

tanh(X) = sinh(X) / cosh(X) , cosh(X) != 0

asinh(X) = ln( X + sqrt(1 + X^2) )

acosh(X) = ln( X + ( sqrt(X + 1) x sqrt(X - 1) ) )

atanh(X) = ( ln(1 + X) - ln(1 - X) ) / 2 , (1+X) != 0 , (1-X) != 0