If you want to be an engineer and physicist, you should consider electrical engineering. A lot of schools offer a double major in physics and ee, since a lot of the topics are similar. Plus with ee, you can get into solid state theory which really combines physics, engineering, and chemistry...
Homework Statement
I need to calculate the residue of a function at infinity. My teacher does this by expanding the function in a laurent expansion and deduces the value from that. That seems much harder than it needs to be. For example, in the notes he calculates the residue at infinity...
Here is a calculation of the residue at z=0.5
http://www.wolframalpha.com/input/?i=residue+4%28z%2Bz^-1%29^2%2F%28-i%28z-2%29%282z-1%29%29+at+z%3D0.5
Here is the integral evaluated
http://www.wolframalpha.com/input/?i=integrate+16cos^2+x+dx%2F%285-4cosx%29+from+0+to+pi
semi circle of radius 1. If you get i in the numerator, when you evaluate the contour integral, wouldn't that give you a negative area? I = 2*i*pi*residue = 2i pi*(25i/3)= -50pi/3
I do, but there is also the factor of i*z from the change of variable in the differential term. So I factor that through the denominator, and it gives me (5z - 2z^2 -2) which factors to (z-2)(2z-1)
The problem has to be with the residue, as that is the only residue which makes up the contour...
Homework Statement
Use the residue theorem to evaluate: \int^{\pi}_{0}\frac{16cos^{2}xdx}{5-4cosx}The Attempt at a Solution
I rewrote the integral with the substitutions
z=e^{ix}
dx = \frac{dz}{iz}
cosx = 0.5(z+z^{-1})
cos^{2}x = 0.25(z+z^{-1})^{2}
I throw all that in, convert the...
wow, sorry I see now. Forgive what I just wrote, it is late and I wasn't thinking correctly.
When I multiply out, I get:
x^{3} -x^{2}(r_{1}+r_{2}+r_{3}) + x(r_{1}r_{2}+r_{1}r_{3}+r_{2}r_{3}) -r_{1}r_{2}r_{3}=0
then you can equate coefficients with the original cubic expression to get...
Is that really all I need to do? I mean when I multiply it out, I get:
x^{3}-cx^{2}-bx^{2}-ax^{2}+bcx+acx+abx-abc=0
The question says to develop formulae, all three formulas are within this equation...is there anything else I can really say here? lol
Homework Statement
Given that the equation x^{3} + ax^{2} + bx + c = 0 has the roots r_{1}, r_{2}, r_{3}, develop formulae for r_{1} + r_{2} + r_{3}, r_{1}r_{2}+r_{1}r_{3}+r_{2}r_{3}, r_{1}r_{2}r_{3}
Homework Equations
The Attempt at a Solution
Not really sure where to...