Recent content by LoA

Homework Statement Compute the first four terms of the Taylor series of \frac{1}{1+e^{z}} at z_{0} = 0 and give it's radius of convergence. Homework Equations e^{z} = \sum\frac{z^{n}}{n!} = 1 + z +\frac{z^{2}}{2!} + \frac{z^{3}}{3!} + o(z^{3}) \frac{1}{1+w} =...

Great, I see what you mean now. I've worked out a rough sketch but I'm still a little uncertain of the algebra. Let me run my sketch past you and then ask how I can make it more concrete. Let z_{0} s.t. |z_{0}| > 0 and arg(z_{0}) = 2\pi . Then f(z_{0}) =...

Wow ok that helps a lot, thanks! So the branch determines what arg(z) is for the log that appears in complex powers? So the multivalued function f(z) = z^1/3 gives three distinct values all the way up to 6pi, and THEN the values begin to repeat? This was a lot simpler when we were just computing...

No, it's close to 2pi/3. So it's roots will be 2pi/3, 4pi/3, 6pi/3=2pi, and this last root will be outside of the selected branch of the cube root function, showing that in the limit the cube root of this guy is not the same as the cube root of a guy on the real axis, or at least doesn't exist...

Ahh ok. So that clarifies my problem arg(z), but it leaves me back at step one. I'm not sure how to show this at all in that case. It seems that since z_{n} approaches z_{0} , the function is continuous on that branch. I apologize if I'm being dense but I'm quite lost as to why this function...

Homework Statement Let f(z) = z^{\frac{1}{3}} be the branch of the cube root function defined on \left[0, 2\pi\right). Show that f is not continuous for z_{0} where z_{0} is such that Re(z_{0}) > 0 and Im(z_{0}) = 0. Homework Equations For a sequence of complex numbers, z_{n} = x_{n} + iy_{n}...

Homework Statement I am confused about what happens to the index of summation when I differentiate a series term by term. Let me show you two examples from my diff eq book (boyce and diprima) which are the primary source of my confusion: Homework Equations From page 268: The function f is...

Homework Statement Let S be a subset of R^{4}: S = {v_{1},v_{2}, v_{3}, v_{4}} = { [1,3,2,0] [-2,0,6,7] [0,6,10,7] [2,10,-3,1] } Determine whether S spans R^{4}. Homework Equations span(S) = {V| V = av_{1}+bv_{2}+cv_{3}+dv_{4}} The Attempt at a Solution When I row...

Homework Statement u^2 --------------- u^2 - 4 Homework Equations I am told that this is equal to 1+ 4/u^2 - 4The Attempt at a Solution No clue how these two are related. Factor out a u^2/u^2? But that alters the denominator. My next that is partial fraction decomp. Am...

Awesome, thank you, I'm really not sure why that didn't click. I think I frequently miss the forest for the trees :redface: BTW, is the notion of a "simplified" related to how many operators have scope over the whole expression? i.e., here it went from two applications of "+" to just one...

Homework Statement The original problem is \int \, \frac {xe^{2x}}{(1+2x)^2} dx . I utilized integration by parts to get: -\frac {xe^{2x}}{2(1+2x)} \, + \, \frac {1}{4}e^{2x} \, + \, C which I know is correct. However, I am told by the book that this may also be expressed as...

Homework Statement A subway train travels over a distance s in t seconds. it starts from rest and ends at rest. In the first part of its journey it moves with constant acceleration f and in the second part with constant deceleration r \, . Show that s \, = \, \frac {[\frac {fr} {f \...