Posted by DarthWho on November 23, 2011 at 11:45 PM | comments (1) |
Sorry for the huge delay in getting this one up my bad.
anyway:
how fo we calculate a^b when both A and B are complex numbers?
well this is where the complex exponentiation formula comes into play (this is a long one so hold onto your hats!
well first we have the formula:
E+Fi=(A+Bi)^(C+Di)
how do we work that out you ask? well it just so happens that that formula is equal to:
E+Fi=(A^2+B^2)^(C/2)*e^(-D*ARG(A+Bi))*(COS(C*ARG(A+Bi)+D*ln(A^2+B^2)/2)+i*SIN(C*ARG(A+Bi)+D*ln(A^2+B^2)/2))
or to split it up into components
G=(A^2+B^2)^(C/2)*e^(-D*ARG(A+Bi))
E=G*COS(C*ARG(A+Bi)+D*ln(A^2+B^2)/2)
F=G*SIN(C*ARG(A+Bi)+D*ln(A^2+B^2)/2)
I will try to update more regularly in the future;
this sunday: Calculus and why programmers should have some basic understanding of the subject. this will kick off a new seried which will focus on educating self professed math dummies on the mysteries of basic calculus.
Posted by DarthWho on May 22, 2011 at 12:00 AM | comments (1) |
the complex argument is a function which is comonly used in functions such as complex exponentiation; it is a function which takes a Complex number and returns a real number to put it into easy to understand terms:
ARG(X + i * Y) = ARCTAN(Y / X)
Or to put it in terms of qb64 code:
ARG(X + i * Y) = ATN(Y / X)
NEXT WEEK: COMPLEX EXPONENTIATION
Posted by DarthWho on May 15, 2011 at 12:00 AM | comments (0) |
Complex numbers are not as difficult as they may first appear to be; this numeric system which is traditionally written in the form of:
A + B * i
Is the second of the division maths to be discovered; this is because like the real numbers the complex numbers are capable of being divided: however the complex numbers could also be written as
(A, B)
which brings us to a rather interesting point where it comes to these complex numbers: they form a twodimensional plane similar to the X,Y plane which is commonly used in programming.
now onto the basic mathematic operations:
Addition
(A + B * i) + (C + D * i) = (A + C) + (B + D) * i
Subtration
(A + B * i) - (C + D * i) = (A - C) + (B - D) * i
Multiplication:
(A + B * i) * (C + D * i) = (A * C - B * D) + (A * D + B * C) * i
and Division:
(A + B * i)/(C + D * i) = ((A * C + B * D) + (B * C + A * D) * i)/(C ^ 2 + d ^ 2)
NEXT WEEK: the Complex Argument. (ARG(z))