Recent content by variety

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    Derivatives of parametric functions (or whatevs)

    Quick question. This is kind of embarrassing actually. Suppose I have functions x(t,s) and y(t,s) (say they're parametric equations of a surface of something) and I want to know what dy/dx is. Specifically, I have x and y in terms of the parameters, which are kind of complicated functions, and I...
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    Counterexample for a premeasure on a semiring over Q

    Homework Statement Let \alpha(r)=r and let P be the family of intervals [a,b) in \mathbb{Q}. Define \mu_{\alpha}([a,b))=\alpha(b)-\alpha(a). Show by example that \mu_{\alpha} is not countably additive.Homework Equations \mu is countably additive if for any sequence of mutually disjoint subsets...
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    Homeomorphism between unit square and unit disc

    Thanks for the help! I got it now.
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    Homeomorphism between unit square and unit disc

    I even have another: f_3(s,t)=\left(\sin{(2\pi s)}\cos{(2\pi t)},\sin{(2\pi s)}\sin{(2\pi t)}\right). However, this is also not injective since all points with s=0 are mapped to the origin.
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    Homeomorphism between unit square and unit disc

    Homework Statement I want to find a bijective function from [0,1] x [0,1] -> D, where D is the closed unit disc. Homework Equations The Attempt at a Solution I have been able to find two continuous surjective functions, but neither is injective. they are...
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    How to prove a topological space is metrizable

    Homework Statement X is a set and P(X) is the discrete topology on X, meaning that P(X) consists of all subsets of X. I want to prove that X is metrizable. Homework Equations My text says that a topological space X is metrizable if it arises from a metric space. This seems a little unclear to...
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    Graduate Different Statements of Morera's Theorem

    In my Complex Analysis text, Morera's theorem states that a continious function whose domain is a region G is analytic if \int_T f=0 for every triangular path T in G. However, other versions of the theorem state that the integral must be zero for any simple closed curve in G. Can someone explain...
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    Finding Roots of Polynomials in C

    Oh I got it. Thanks a lot!
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    Finding Roots of Polynomials in C

    Homework Statement Show the the equation f(z) = z^4 + iz^2 + 2 = 0 has two roots with |z|=1 and two roots with |z|=sqrt(2), without actually solving the equation.Homework Equations Rouche's theorem, the argument principle?The Attempt at a Solution This is what I have done so far: First show...
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    Graduate Is Any Closed Curve Homotopic to a Constant Curve in a Simply Connected Region?

    I have a question about the definition of null homotopic. My textbook (Functions of one Complex Variable I by John B. Conway) defines it as follows: If \gamma is a closed rectifiable curve in a region G, then \gamma is homotopic to zero if \gamma is homotopic to a constant curve. My question is...
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    Entire Function with Negative Imaginary Values: Proving Constantness

    Homework Statement Let f:C->C be an entire function such that Imf(z) <= 0 for all z in C. Prove that f is constant. Homework Equations Cauchy-Riemann equations?? The Attempt at a Solution I don't know why I haven't been able to get anywhere with this problem. I feel like I have to...
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    Undergrad Absolute value of complex numbers

    What I'm asking is, are there any conditions on a and b such that the above statement is true?
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    Undergrad Absolute value of complex numbers

    When is it true that is |a|=|b|, then either a=b or a=b*, where a and b are complex numbers and b* is the complex conjugate of b?
    • variety
    • Thread
    • Absolute Absolute value Complex Complex numbers Numbers Value
    • Replies: 3
    • Forum: Calculus
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    Graduate Continuity of complex functions

    Do you guys know of any functions which are continuous on the real line, but discontinuous on the complex plane? If not, is there a reason why this can never happen?
    • variety
    • Thread
    • Complex Continuity Functions
    • Replies: 3
    • Forum: Calculus
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    Graduate Uncountability of neighborhoods in R^n

    Yeah sorry I wasn't that clear. I just want to know if the set {x in R : |x-y|<r} is uncountable.