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Math Help - Continuity of a complex-valued function

  1. #1
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    Continuity of a complex-valued function

    Define the function f(z)=z*Real Part(z)/|z|,z!= 0
    0 ,z=0
    Prove that f(z) is continuous in the entire complex plane.

    Pls somebody help me with dis one......thanx in advance.
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  2. #2
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    Quote Originally Posted by deepakpc007 View Post
    Define the function f(z)=z*Real Part(z)/|z|,z!= 0
    0 ,z=0
    Prove that f(z) is continuous in the entire complex plane.
    If z=re^{\mathif{i}\theta} then \frac{z\cdot\text{Re}(z)}{|z|}=\frac{(re^{\mathif{  i}\theta})(r\cos(\theta)}{r}.

    Now what happens when r\to 0~?
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    Quote Originally Posted by Plato View Post
    If z=re^{\mathif{i}\theta} then \frac{z\cdot\text{Re}(z)}{|z|}=\frac{(re^{\mathif{  i}\theta})(r\cos(\theta)}{r}.

    Now what happens when r\to 0~?
    when r\to 0~?, z=re^{\mathif{i}\theta} then \frac{z\cdot\text{Re}(z)}{|z|}=\frac{(re^{\mathif{  i}\theta})(r\cos(\theta)}{r}=infinity???I think so.
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    Quote Originally Posted by deepakpc007 View Post
    when r\to 0~?, z=re^{\mathif{i}\theta} then \frac{z\cdot\text{Re}(z)}{|z|}=\frac{(re^{\mathif{  i}\theta})(r\cos(\theta)}{r}=infinity???I think so.
    Oh my no.
    Is this true: r\not= 0 so \frac{(re^{\mathif{i}\theta})(r\cos(\theta)}{r}=(r  e^{\mathif{i}\theta})(\cos(\theta))~?
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  5. #5
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    Quote Originally Posted by Plato View Post
    Oh my no.
    Is this true: r\not= 0 so \frac{(re^{\mathif{i}\theta})(r\cos(\theta)}{r}=(r  e^{\mathif{i}\theta})(\cos(\theta))~?
    Yes thats true.How can i prove with this?
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  6. #6
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    Quote Originally Posted by deepakpc007 View Post
    Yes thats true.How can i prove with this?
    What is there to prove.
    As r\to 0 the other factors are bounded, so what is the limit?
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  7. #7
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    Quote Originally Posted by Plato View Post
    What is there to prove.
    As r\to 0 the other factors are bounded, so what is the limit?
    Sorry sir,i dont know.i am not dat good in maths.please help me.

    Edit : As r\to 0 the other factors are bounded,means a finite limit.Is it so?
    Limit is 0 and hence the function is continuous in the entire complex plane.Am i right?
    Last edited by deepakpc007; April 29th 2011 at 05:53 AM.
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