# Search results for query: *

1. ### Convert Partial Fractions & Taylor Series: Solving Complex Equations

$$\frac{1}{z}\left(\frac{1}{z-2} - \frac{1}{z-1}\right)= \frac{1}{z}\left(\frac{1}{-2}\cdot\frac{1}{1-\frac{z}{2}} - \frac{1}{z}\cdot\frac{1}{1-\frac{1}{z}}\right)$$ For ##1<|z|<2##, we have, ##\left|\frac{1}{z}\right|<1## and ##\left|\frac{z}{2}\right|<1##. So you can now use the formula for...
2. ### Complex Integration using residue theorem

Homework Statement [/B] ##C_\rho## is a semicircle of radius ##\rho## in the upper-half plane. What is $$\lim_{\rho\rightarrow 0} \int_{C_{\rho}} \frac{e^{iaz}-e^{ibz}}{z^2} \,dz$$ Homework Equations If ##C## is a closed loop and ##z_1, z_2 ... z_n## are the singular points inside ##C##...
3. ### How integrate 1/(x(1+x^2)^0.5) dx

Substitute ##x=\tan{y}##.
4. ### Find the limit of the function

$$e^{-t^2}\leq1$$ $$\int^x_0e^{-t^2}\,dt\leq\int^x_0\,dt$$ $$\frac{\int^x_0e^{-t^2}\,dt}{x}\leq\frac{\int^x_0\,dt}{x}$$
5. ### Find the limit of the function

Thanks for pointing out the mistake. I did not mean to analytically calculate the area, I meant to use the trapezoidal approximation for the area.
6. ### Find the limit of the function

Look, if you put ##x=0##, you have ##\frac{0}{0}## form [##\int_0^x e^{-t^2}dt## is zero when ##x=0##]. So in that case, you cannot conclude that the limit is infinity. Now, look at the figure: If ##x\rightarrow 0##, what will be ##\frac{\int_0^x e^{-t^2}dt}{x}=\frac{Area}{x}## ? [Use...
7. ### Problem about an Eigenvector

Homework Statement If ##AA^T=A^TA##, then prove that ##A## and ##A^T## have the same eigenvectors. Homework Equations The Attempt at a Solution ##Ax=\lambda x## ##A^TAx=\lambda A^Tx## ##AA^Tx=\lambda A^Tx## ##A(A^Tx)=\lambda (A^Tx)## So, ##A^Tx## is also an eigenvector of ##A##. What should...