Edit: when I look at it more closely Im confused myself... is t a real variable?
I know that in a complex number such as
"Re" is supposed to be the "real part", which is so that and that "Im" is supposed to be the "imaginary part", which is so that in the above example.
However, recently I stumpled upon a problem when trying to solve linear systems:
I'm not sure that I understand how exactly to execute this formula with the "Re" part.
Here is an example:
Can somebody explain to me what the "Re" did here?
And what about "Im"? If it had said "Im" instead, what would be different in the answer?
in general, for a complex number z:
so if , then .
for example:
.
what you want to do, if you have complex expressions multiplied together, is first "multiply them out", to get something in the form a+bi.
EDIT: i suspect what you actually WANT is:
parentheses matter!!!!
if so then the real part is:
Yes, I have looked at it 5 times now and my book writes this as a solution. Here it is in its entirety:
A linear system is given as:
And
The transfer function is calculated to and given that then a linear system has the solution:
Where
Since in this example, we have the formula for the solution:
Replacing "s" with the imaginary unit in the transfer function:
Inserting into the solution gives:
Using Euler's formula we now have:
Which according to my book:
see my post above. you are missing parentheses. people are thinking you mean the fraction is just times the cosine term, instead of the entire cosine + imaginary sine expression.
what you have are two complex numbers multiplied together.
now (a+bi)(c+di) = (ac - bd) + (ad + bc)i, and the real part of this is ac - bd.
Yes that is true about the parentheses. But I'm still not sure I understand the last part.
So we have that [TEX]\frac{-3+4i}{25}[TEX] where the 4i is the imaginary part, so that disappears, leaving:
But then I don't understand where the fraction comes from. Isn't 4i the imaginary part?
What is the multiplication rule that you mention? I haven't seen that before.
I just used the and that worked out really well!
But I've never seen this rule before? :S
Also is there a similar rule for Im?
Okay, is this correct then?
I think I got it now?
it's very simple.
to find the real part of a complex number easily, you have to have it in the form:
(something real) + i (something else that's real).
now what you have is a complex fraction (a+bi)/d. that is the complex number (a/d) + (b/d)i.
you have it multiplied by another complex number: cos(t) + i sin(t).
when we multiply these together:
[(a+bi)/d](cos(t) + i sin(t)), we get:
(a/d)cos(t) - (b/d)sin(t) + [(a/d)sin(t) + (b/d)cos(t)]i
the real part is thus: (a/d)cos(t) - (b/d)sin(t) and the imaginary part is:
(a/d)sin(t) + (b/d)cos(t)
in your example above:
a = -3
b = 4
d = 25
yes, the imaginary part of (a+bi)(c+di) is: ad+bc. why? this is the very DEFINITION of complex multiplication:
(a+bi)(c+di) = a(c+di) + (bi)(c+di) = ac + (ad)i + (bi)c + (bi)(di) = ac + (ad)i + (bc)i + (bd)i^{2}.
but i^{2} = -1, so:
ac + (ad)i + (bc)i + (bd)i^{2} = ac + (ad)i + (bc)i + bd(-1) = ac - bd + (ad + bc)i (collecting the "i" terms).