Recent content by fauboca

Let ##\gamma## be a closed curve in ##\mathbb{C}##. If ##\gamma## doesn't contain any point from [2,5] in its interior, then ##\int_{\gamma}f=0## since f is holomorphic away from [2,5]. Suppose that ##\gamma## contains [2,5] in its interior. Let R be a rectangle oriented with the coordinate...

Shouldn't the square contain the interval? So it would be (1.9,-.1), (1.9, .1), (5.1,-.1),(5.1,.1)?

For your solution when k=1, why do you have [-1,\infty)? If t = -1, you are dividing by zero. What is wrong with the (-\infty, -1)? (-2)^2-4+1>0 Is this true for the entire set?

I have this proof for finite points but how would I modify it for infinite many points between [2,5]? Assume q(z) is any function that is holomorphic on a disc U except at a finite number of points \xi_1,\ldots, \xi_n\in U, and assume \lim_{z\to\xi_j}(z-\xi_j)q(z)=0 for 1\leq j\leq n. Let...

Isn't this by definition. f is continuous on a compact set so that it is uniformly continuous.

So I am trying to use Morera's Theorem: Let U be an open set in C and let f be continuous on U. Assume that the integral of f along the boundary of every closed rectangle in U is 0. Then f is holomorphic. So let U = \mathbb{C} - [2,5] Let R be rectangles in U which are parallel to the...

If you set I = integral and multiplied by the same integral, you would have I^2. When you solve that you get 2pi but then you take the square root. However, isn't is supposed to be equal to \frac{1}{\sqrt{2\pi}

Eigenspace are the eigenvectors. I obtained a different eigenvector for you second one. I don't believe I made a mistake but I could have.

Ok thanks.

What can be done to justify slipping differentiation past the integral? How can I show the partials are continuous at this point?

How about something easier? Is R compact? Is the unit circle compact? Can you have continuous map from a compact set to a non compact set?

The Taylor Series expansion of f(z) = \sum_{n = 0}^{\infty}c_n(z - z_0)^n = c_0 + c_1(z - z_0) + c_2(z - z_0)^2 +\cdots, and f(z_0) = c_0. So, $$ f(z) - f(z_0) = c_1(z - z_0) + c_2(z - z_0)^2 +\cdots. $$ By factoring, we obtain f(z) - f(z_0) = c_1(z - z_0)\left[1 + \frac{c_2}{c_1}(z -...

I am still lost on how to do this though.

Separate to log = pi i/2 and raise use the inverse function e.

There may be a better way to do this one. Let me think.