Recent content by Pyroadept

Homework Statement State, with justification, if the Fundamental Theorem of Contour Integration can be applied to the following integrals. Evaluate both integrals. \begin{eqnarray*} (i) \hspace{0.2cm} \int_\gamma \frac{1}{z} dz \\ (ii) \hspace{0.2cm} \int_\gamma \overline{z} dz \\...

Ah, because an open interval can be written as the countably infinite intersection of closed intervals... Hmm, ok, let me think about it some more :) Thanks for your help!

Hi Krylov/Andrew, If I take the (finite) union of n number of intervals on the real line in the form (-∞,a], then the resultant union is going to be in the form (-∞,a], yes? Because there will always be some a' that is bigger than all the other a's, hence it will be the right-most interval for...

Thanks for your response, Andrew! a counterexample - in this case a countable union that is not in the set: To get this, as the outcome we would have to have some set that was in the form (-∞,a) (i.e. right interval is open) but that would mean there is some set in our union of this form, which...

Homework Statement Let ε = { (-∞,a] : a∈ℝ } be the collection of all intervals of the form (-∞,a] = {x∈ℝ : x≤a} for some a∈ℝ. Is ε closed under countable unions? Homework Equations Potentially De Morgan's laws? The Attempt at a Solution Hi everyone, Thanks in advance for looking at my...

Thanks guys, I appreciate it! :)

I thought it might come out looking like an 'In'... :) So if I get it down to: p^{a}(1-p)^{b} = e^c how can I solve for p then? Assuming the a and b are large-ish numbers like, say, 10 and 20.

Homework Statement (Here, by 'log' I mean natural logarithm) Solve for x: a.log(x) + b.log(1-x) = c for a, b and c constants Homework Equations The Attempt at a Solution Hi everyone, This is so embarrassing but this is really stumping me! I know how to do it if a=b...

Actually, I have had a wave of inspiration since - is this correct? The singularities occur for 2.cos(z)-1 = 0 i.e. cos(z) = 1/2 This happens for z = pi/3 (+ 2k.pi, but this z is the smallest one) So then the distance from z=1 to z=pi/3 is: √(1 - pi/3)^2) = 2pi/3 which is then...

Homework Statement Find the radius of convergence of the Taylor series at 0 of this function f(z) = \frac{e^{z}}{2cosz-1} Homework Equations The Attempt at a Solution Hi everyone, Here's what I've done so far: First, I tried to re-write it as a Laurent series to find...

Thanks guys, I appreciate it :)

Also, just out of interest, how exactly would you write that formula as a power/Laurent series?

Thanks for your reply! Ah. So the singularity would be at z=0 then? So the distance from z=0 to z=1 would be 1, so the radius of convergence is then R=1 centered at the point z=1?

Homework Statement Find the radius of convergence of the Taylor series at z = 1 of the function: \frac{1}{e^{z}-1} Homework Equations The Attempt at a Solution Hi everyone, Here's what I've done so far. Multiply top and bottom by minus 1 to get: -1/(1-e^z) And then...

Thanks!