what is the mapping of horizontal lines and vertical lines under $w(z)=sin(z)$ in general?












0












$begingroup$


For the mapping of horizontal lines ($z=x+iy_0$ for fixed $y_0$) and vertical lines ($z=x_0+iy$ for fixed $x_0$) under $w(z) = sin(z)$, are there any general formulas?



What I mean is, is there a general formula like "$z=x_0+iy$ is mapped to a hyperbola with the vertex (something in terms of $x_0$) and the distant from the origin to the vertex (something in terms of $x_0$)"?



I used $$sin(x_0+iy) = sin(x_0) cosh(y) +i cos(x_0) sinh(y)$$ and $$ sin(x+iy_0) = sin(x) cosh(y_0) + i cos(x) sinh(y_0)$$, but I don't know how to go beyond that to obtain a general formula for the mapping under $w=sin(z)$.



Thank you in advance.






complex-analysis complex-numbers analytic-functions







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  • $begingroup$
    Have you tried the addition formula for the sine function?
    $endgroup$
    – saulspatz
    Dec 22 '18 at 20:05










  • $begingroup$
    @saulspatz Yes, I did, but I still don't understand how to derive a general formula for mapping under sin(z).
    $endgroup$
    – Duc Van Khanh Tran
    Dec 22 '18 at 20:54






  • 1




    $begingroup$
    Welcome to MSE. You'll get a lot more help if you explain what you did, how far you got, and where you're stuck in the body of your question. You'll find that questions that say, "Here's a problem I can't solve; please do it for me," usually don't attract much of a response. Please be sure to respond by editing the question body. Many people browsing questions will vote to close without looking at the comments.
    $endgroup$
    – saulspatz
    Dec 22 '18 at 20:57
















0












$begingroup$


For the mapping of horizontal lines ($z=x+iy_0$ for fixed $y_0$) and vertical lines ($z=x_0+iy$ for fixed $x_0$) under $w(z) = sin(z)$, are there any general formulas?



What I mean is, is there a general formula like "$z=x_0+iy$ is mapped to a hyperbola with the vertex (something in terms of $x_0$) and the distant from the origin to the vertex (something in terms of $x_0$)"?



I used $$sin(x_0+iy) = sin(x_0) cosh(y) +i cos(x_0) sinh(y)$$ and $$ sin(x+iy_0) = sin(x) cosh(y_0) + i cos(x) sinh(y_0)$$, but I don't know how to go beyond that to obtain a general formula for the mapping under $w=sin(z)$.



Thank you in advance.






complex-analysis complex-numbers analytic-functions







share|cite|improve this question











$endgroup$












  • $begingroup$
    Have you tried the addition formula for the sine function?
    $endgroup$
    – saulspatz
    Dec 22 '18 at 20:05










  • $begingroup$
    @saulspatz Yes, I did, but I still don't understand how to derive a general formula for mapping under sin(z).
    $endgroup$
    – Duc Van Khanh Tran
    Dec 22 '18 at 20:54






  • 1




    $begingroup$
    Welcome to MSE. You'll get a lot more help if you explain what you did, how far you got, and where you're stuck in the body of your question. You'll find that questions that say, "Here's a problem I can't solve; please do it for me," usually don't attract much of a response. Please be sure to respond by editing the question body. Many people browsing questions will vote to close without looking at the comments.
    $endgroup$
    – saulspatz
    Dec 22 '18 at 20:57














0












0








0





$begingroup$


For the mapping of horizontal lines ($z=x+iy_0$ for fixed $y_0$) and vertical lines ($z=x_0+iy$ for fixed $x_0$) under $w(z) = sin(z)$, are there any general formulas?



What I mean is, is there a general formula like "$z=x_0+iy$ is mapped to a hyperbola with the vertex (something in terms of $x_0$) and the distant from the origin to the vertex (something in terms of $x_0$)"?



I used $$sin(x_0+iy) = sin(x_0) cosh(y) +i cos(x_0) sinh(y)$$ and $$ sin(x+iy_0) = sin(x) cosh(y_0) + i cos(x) sinh(y_0)$$, but I don't know how to go beyond that to obtain a general formula for the mapping under $w=sin(z)$.



Thank you in advance.






complex-analysis complex-numbers analytic-functions







share|cite|improve this question











$endgroup$




For the mapping of horizontal lines ($z=x+iy_0$ for fixed $y_0$) and vertical lines ($z=x_0+iy$ for fixed $x_0$) under $w(z) = sin(z)$, are there any general formulas?



What I mean is, is there a general formula like "$z=x_0+iy$ is mapped to a hyperbola with the vertex (something in terms of $x_0$) and the distant from the origin to the vertex (something in terms of $x_0$)"?



I used $$sin(x_0+iy) = sin(x_0) cosh(y) +i cos(x_0) sinh(y)$$ and $$ sin(x+iy_0) = sin(x) cosh(y_0) + i cos(x) sinh(y_0)$$, but I don't know how to go beyond that to obtain a general formula for the mapping under $w=sin(z)$.



Thank you in advance.






complex-analysis complex-numbers analytic-functions




complex-analysis complex-numbers analytic-functions






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 25 '18 at 11:15









Lee David Chung Lin

4,50841342




4,50841342










asked Dec 22 '18 at 20:04









Duc Van Khanh TranDuc Van Khanh Tran

44




44












  • $begingroup$
    Have you tried the addition formula for the sine function?
    $endgroup$
    – saulspatz
    Dec 22 '18 at 20:05










  • $begingroup$
    @saulspatz Yes, I did, but I still don't understand how to derive a general formula for mapping under sin(z).
    $endgroup$
    – Duc Van Khanh Tran
    Dec 22 '18 at 20:54






  • 1




    $begingroup$
    Welcome to MSE. You'll get a lot more help if you explain what you did, how far you got, and where you're stuck in the body of your question. You'll find that questions that say, "Here's a problem I can't solve; please do it for me," usually don't attract much of a response. Please be sure to respond by editing the question body. Many people browsing questions will vote to close without looking at the comments.
    $endgroup$
    – saulspatz
    Dec 22 '18 at 20:57


















  • $begingroup$
    Have you tried the addition formula for the sine function?
    $endgroup$
    – saulspatz
    Dec 22 '18 at 20:05










  • $begingroup$
    @saulspatz Yes, I did, but I still don't understand how to derive a general formula for mapping under sin(z).
    $endgroup$
    – Duc Van Khanh Tran
    Dec 22 '18 at 20:54






  • 1




    $begingroup$
    Welcome to MSE. You'll get a lot more help if you explain what you did, how far you got, and where you're stuck in the body of your question. You'll find that questions that say, "Here's a problem I can't solve; please do it for me," usually don't attract much of a response. Please be sure to respond by editing the question body. Many people browsing questions will vote to close without looking at the comments.
    $endgroup$
    – saulspatz
    Dec 22 '18 at 20:57
















$begingroup$
Have you tried the addition formula for the sine function?
$endgroup$
– saulspatz
Dec 22 '18 at 20:05




$begingroup$
Have you tried the addition formula for the sine function?
$endgroup$
– saulspatz
Dec 22 '18 at 20:05












$begingroup$
@saulspatz Yes, I did, but I still don't understand how to derive a general formula for mapping under sin(z).
$endgroup$
– Duc Van Khanh Tran
Dec 22 '18 at 20:54




$begingroup$
@saulspatz Yes, I did, but I still don't understand how to derive a general formula for mapping under sin(z).
$endgroup$
– Duc Van Khanh Tran
Dec 22 '18 at 20:54




1




1




$begingroup$
Welcome to MSE. You'll get a lot more help if you explain what you did, how far you got, and where you're stuck in the body of your question. You'll find that questions that say, "Here's a problem I can't solve; please do it for me," usually don't attract much of a response. Please be sure to respond by editing the question body. Many people browsing questions will vote to close without looking at the comments.
$endgroup$
– saulspatz
Dec 22 '18 at 20:57




$begingroup$
Welcome to MSE. You'll get a lot more help if you explain what you did, how far you got, and where you're stuck in the body of your question. You'll find that questions that say, "Here's a problem I can't solve; please do it for me," usually don't attract much of a response. Please be sure to respond by editing the question body. Many people browsing questions will vote to close without looking at the comments.
$endgroup$
– saulspatz
Dec 22 '18 at 20:57










1 Answer
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$begingroup$

Let $w = u + iv = sin(x)cosh(y) + icos(x)sinh(y)$



For $y = y_0$ we have the parametrization



begin{align}
u(x) &= asin(x) \
v(x) &= bcos(x)
end{align}



where $a=cosh(y_0)$ and $b=sinh(y_0)$. This represents a horizontal ellipse with major and minor axes $a$ and $b$, respectively, since $frac{u^2}{a^2}+frac{v^2}{b^2}=1$. All ellipses have the focus $sqrt{a^2-b^2}=1$



Similarly, for $x=x_0$ we have the paremetrization



begin{align}
u(y) &= acosh(y) \
v(y) &= bsinh(y)
end{align}



where $a = sin(x_0)$ and $b = cos(x_0)$. This represents a hyperbola with major axis $a$, since $frac{u^2}{a^2} - frac{v^2}{b^2}=1$. The focus of every hyperbola is also $sqrt{a^2+b^2}=1$






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    0












    $begingroup$

    Let $w = u + iv = sin(x)cosh(y) + icos(x)sinh(y)$



    For $y = y_0$ we have the parametrization



    begin{align}
    u(x) &= asin(x) \
    v(x) &= bcos(x)
    end{align}



    where $a=cosh(y_0)$ and $b=sinh(y_0)$. This represents a horizontal ellipse with major and minor axes $a$ and $b$, respectively, since $frac{u^2}{a^2}+frac{v^2}{b^2}=1$. All ellipses have the focus $sqrt{a^2-b^2}=1$



    Similarly, for $x=x_0$ we have the paremetrization



    begin{align}
    u(y) &= acosh(y) \
    v(y) &= bsinh(y)
    end{align}



    where $a = sin(x_0)$ and $b = cos(x_0)$. This represents a hyperbola with major axis $a$, since $frac{u^2}{a^2} - frac{v^2}{b^2}=1$. The focus of every hyperbola is also $sqrt{a^2+b^2}=1$






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      Let $w = u + iv = sin(x)cosh(y) + icos(x)sinh(y)$



      For $y = y_0$ we have the parametrization



      begin{align}
      u(x) &= asin(x) \
      v(x) &= bcos(x)
      end{align}



      where $a=cosh(y_0)$ and $b=sinh(y_0)$. This represents a horizontal ellipse with major and minor axes $a$ and $b$, respectively, since $frac{u^2}{a^2}+frac{v^2}{b^2}=1$. All ellipses have the focus $sqrt{a^2-b^2}=1$



      Similarly, for $x=x_0$ we have the paremetrization



      begin{align}
      u(y) &= acosh(y) \
      v(y) &= bsinh(y)
      end{align}



      where $a = sin(x_0)$ and $b = cos(x_0)$. This represents a hyperbola with major axis $a$, since $frac{u^2}{a^2} - frac{v^2}{b^2}=1$. The focus of every hyperbola is also $sqrt{a^2+b^2}=1$






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        Let $w = u + iv = sin(x)cosh(y) + icos(x)sinh(y)$



        For $y = y_0$ we have the parametrization



        begin{align}
        u(x) &= asin(x) \
        v(x) &= bcos(x)
        end{align}



        where $a=cosh(y_0)$ and $b=sinh(y_0)$. This represents a horizontal ellipse with major and minor axes $a$ and $b$, respectively, since $frac{u^2}{a^2}+frac{v^2}{b^2}=1$. All ellipses have the focus $sqrt{a^2-b^2}=1$



        Similarly, for $x=x_0$ we have the paremetrization



        begin{align}
        u(y) &= acosh(y) \
        v(y) &= bsinh(y)
        end{align}



        where $a = sin(x_0)$ and $b = cos(x_0)$. This represents a hyperbola with major axis $a$, since $frac{u^2}{a^2} - frac{v^2}{b^2}=1$. The focus of every hyperbola is also $sqrt{a^2+b^2}=1$






        share|cite|improve this answer









        $endgroup$



        Let $w = u + iv = sin(x)cosh(y) + icos(x)sinh(y)$



        For $y = y_0$ we have the parametrization



        begin{align}
        u(x) &= asin(x) \
        v(x) &= bcos(x)
        end{align}



        where $a=cosh(y_0)$ and $b=sinh(y_0)$. This represents a horizontal ellipse with major and minor axes $a$ and $b$, respectively, since $frac{u^2}{a^2}+frac{v^2}{b^2}=1$. All ellipses have the focus $sqrt{a^2-b^2}=1$



        Similarly, for $x=x_0$ we have the paremetrization



        begin{align}
        u(y) &= acosh(y) \
        v(y) &= bsinh(y)
        end{align}



        where $a = sin(x_0)$ and $b = cos(x_0)$. This represents a hyperbola with major axis $a$, since $frac{u^2}{a^2} - frac{v^2}{b^2}=1$. The focus of every hyperbola is also $sqrt{a^2+b^2}=1$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 23 '18 at 10:59









        DylanDylan

        14.4k31127




        14.4k31127






























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