Search results for query: *

$$\frac{1}{z}\left(\frac{1}{z-2} - \frac{1}{z-1}\right)= \frac{1}{z}\left(\frac{1}{-2}\cdot\frac{1}{1-\frac{z}{2}} - \frac{1}{z}\cdot\frac{1}{1-\frac{1}{z}}\right)$$ 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...

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##...

Substitute ##x=\tan{y}##.

$$e^{-t^2}\leq1$$ $$\int^x_0e^{-t^2}\,dt\leq\int^x_0\,dt$$ $$\frac{\int^x_0e^{-t^2}\,dt}{x}\leq\frac{\int^x_0\,dt}{x}$$

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.

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...

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...

Homework Statement Prove that, $$\sum _{n=1,3,5...} \frac{1}{n} e^{-nx} \sin{ny} = \frac{1}{2}\tan^{-1} (\frac{\sin{y}}{\sinh{x}})$$ Homework Equations $$\tan^{-1}{x} = x - \frac{x^3}{3} +\frac{x^5}{5} - ... $$ 3. The Attempt at a Solution $$\sum _{n=1,3,5...} \frac{1}{n} e^{-nx}...

Homework Statement Let ##u## and ##v## be differentiable functions of ##x,~y## and ##z##. Show that a necessary and sufficient condition that ##u## and ##v## are functionally related by the equation ##F(u,v)=0## is that ##\vec \nabla u \times \vec \nabla v= \vec 0## Homework Equations (Not...

Thanks for your comments. This was an example from 'Mathematical Methods for Physics and Engineering' by Riley, Hobson and Bence (page 133). This is a very reliable book. I think I lack some important concept here.

Homework Statement ##P(z) = 1 - \frac{z}{2} + \frac{z^2}{4} - \frac{z^3}{8} + ... ## Determine if the series is convergent or divergent if ## |z| = 2 ##, where, ## z## is a complex number. Homework Equations ##1+r+r^2+r^3+...+r^{N-1}=\frac{1-r^N}{1-r}## The Attempt at a Solution Let, ##z = 2...

Homework Statement $$\lim _{x \rightarrow 1} (\frac{23}{1-x^{23}}-\frac{11}{1-x^{11}})$$ Homework Equations i) For functions f and g which are differentiable on an open interval I except possibly at a point c contained in I, if , and exists, and for all x in I with x ≠ c, then...

Homework Statement $$ \sum_{n=1}^\infty\frac{1}{1+(a+nb)^2} = ? $$ 2. The attempt at a solution I approximated the result by integration, $$ \begin{align} \sum_{n=1}^\infty \frac{1}{1+(a+nb)^2} &\approx \lim_{N \rightarrow +\infty} {\int_{0}^N \frac{1}{1+(a+bx)^2} dx}\\ &= \lim_{N...