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    Holomorphic on C

    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...
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    Holomorphic on C

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

    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?
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    Holomorphic on C

    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...
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    Bounded continous implies uniformly continuous

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

    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...
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    Improper integral

    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}
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    Eigenvalues, eigenvectors, and eigenspaces

    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.
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    Fourier Transform

    What can be done to justify slipping differentiation past the integral? How can I show the partials are continuous at this point?
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    [Cardinality] Prove there is no bijection between two sets

    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?
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    Cauchy Integral Formula application

    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 -...
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    Holomorphic on C

    I am still lost on how to do this though.
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    Complex Variables Algebra Solutions / Argument/Modulus

    Separate to log = pi i/2 and raise use the inverse function e.
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    Complex Variables Algebra Solutions / Argument/Modulus

    There may be a better way to do this one. Let me think.
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    Complex Variables Algebra Solutions / Argument/Modulus

    so a = x+yi. Then (x+yi)^2-1 = (x^2-y^2-1)+2xyi. What is the modulus of that? Put your i's together as well.
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    Adjoint of Linear Operator

    Then why did you ask if you knew that part? What you have as a proof doesn't look like you used the polar identity. I am not really sure what you have done to claim <T(x),T(y)>=<x,y>
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    Adjoint of Linear Operator

    Because once you prove ||T(x)|| = ||x||, what does ||T(x+y)|| = ??
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    Adjoint of Linear Operator

    Your question says, "Let T be a linear operator..." What does that mean?
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    Adjoint of Linear Operator

    If you use the polar identity, your next step should be \frac{1}{4}\left(||T(x+y)||^2-||T(x-y)||^2\right) =\frac{1}{4}\left(||x+y||^2-||x-y||^2\right)=\langle x,y\rangle
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    Group Theory -

    Have you tried going from what you have and then using the first iso theorem?
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    Adjoint of Linear Operator

    Here is a hint: \langle T(x),T(y)\rangle = \frac{1}{4}\left(||T(x)+T(y)||^2-||T(x)-T(y)||^2\right)=\cdots That is by the polarization identity.
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    Number Theory Question 2

    It seems ok but could be written better. Like you said m is a product of primes so lets write m as m=p_1^{\alpha_1}p_2^{\alpha_2}\ldots p_t^{\alpha_t}. Then m^n = \left(p_1^{\alpha_1}p_2^{\alpha_2}\ldots p_t^{\alpha_t}\right)^n = p_1^{n\alpha_1}p_2^{n\alpha_2}\ldots p_t^{n\alpha_t}. Since...
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    If f is meromorphic on U with only a finite number of poles, then

    If the pole is at z_0, then f=\frac{g}{(z-z_0)} Let S be the set of the poles of f in U and let h be analytic in U with singularities at the points in S. Let the order of the singularity be the order of the pole in f. Then fh has only removable singularities in U. fh=g\Rightarrow f=\frac{g}{h}
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    How many ways are there to solve this question?

    If a is a parameter, you shouldn't name the vector a as well. Have you read the definition coplanar?
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    If f is meromorphic on U with only a finite number of poles, then

    If f is meromorphic on U with only a finite number of poles, then f=\frac{g}{h} where g and h are analytic on U. We say f is meromorphic, then f is defined on U except at discrete set of points S which are poles. If z_0 is such a point, then there exist m in integers such that (z-z_0)^mf(z)...
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    Derivative exponential problem

    I don't get either of your answers when I take the derivative. (1-2x)e^{-x} = e^{-x} -2xe^{-x} = -e^{-x} -2(product rule) = ??
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    Find the necessary and sufficient conditions on the real numbers a,b,c

    What you know now is a must be 0. The question is can b and c be anything?
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    Adjoint of Linear Operator

    Look up isometries it should help you with this problem.
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    Find the necessary and sufficient conditions on the real numbers a,b,c

    Instead of dividing out the 1st row by a, leave it as a. You will still have a = 0. Why is c = 0? the second row means x_3 = 0.