Homework Statement
Find the complete asymptotic expansion of the function \sqrt{1+x^{6}} - x^{3}lnx as x\rightarrow\infty and x\rightarrow0.
In each case give the asymptotic sequence in decreasing orders of magnitude.
Homework Equations
I tried the Taylor Expansion about 0. But I don't...
Homework Statement
\int^{2\pi}_{0}cos^{2}(\theta)sin^{2}(\theta)cos(\theta)sin(\theta)d\theta
If I set x=cos^{2}(\theta), the integral limit should be from 1 to 0 or need I break this integral into to 4 parts (i.e from 1 to 0 plus from 0 to 1 plus from 1 to 0 plus from 0 to 1)?
Homework...
\int^{\pi}_{0}Sin(2Cos(\theta))Cos(2n\theta)d\theta
I can apply some software to do this integral. However, I need some procedures for this integral. Any help is welcome
The original problem is this:
\oint\frac{(z-a)e^{z}}{(z+a)sinz}dz c=2a centered at z=0 2a<pi
we can express the integral around the contour as the sum of the integral around z1 and z2 where the contour is a small circle around each pole. Call these contours C1 around z1 and C2 around z2...
yes.
I have done the form like this:
\oint\frac{(z-a)e^{z}}{(z+a)}\frac{dz}{sinz} + \oint\frac{(z-a)e^{z}}{sinz}\frac{dz}{(z+a)}
however the first one is not the standard Cauchy Integral Formula
Homework Statement
Using the Cauchy Integral Formula compute the following integrals,where C is a circle of radius 2a centered at z=o, where 2a<pi
Homework Equations
\oint\frac{(z-a)e^{z}}{(z+a)sinz}
The Attempt at a Solution
Homework Statement
Consider a pendulum oscillation problem, where pendulum oscillates around the vertical in the downward configuration.
Assume that there is no friction at the pivot point around which the pendulum rotates, and assume that there exists a torsional spring that counter acts...