How to Write Complex Functions in u+iv Form

Step-by-Step Guide

1. 1

   Step 1: Write w=ez{\displaystyle w=e^{z}} in terms of its real and imaginary components.

   The exponential function is one of the first functions introduced in complex analysis for many reasons, most notably being that it is its own derivative, and highlights the very important relationship between rotations and exponentials. , Use the exponent relation ea+b=eaeb.{\displaystyle e^{a+b}=e^{a}e^{b}.} w=ex+iy=exeiy{\displaystyle w=e^{x+iy}=e^{x}e^{iy}} , w=ex(cos⁡y+isin⁡y){\displaystyle w=e^{x}(\cos y+i\sin y)} The function is now in u+iv{\displaystyle u+iv} form.
   
   Here, we have u(x,y)=excos⁡y{\displaystyle u(x,y)=e^{x}\cos y} and v(x,y)=exsin⁡y.{\displaystyle v(x,y)=e^{x}\sin y.} , Substitute z=x+iy{\displaystyle z=x+iy} into the function. w=1x+iy{\displaystyle w={\frac {1}{x+iy}}} , 1x+iyx−iyx−iy=x−iyx2+y2{\displaystyle {\frac {1}{x+iy}}{\frac {x-iy}{x-iy}}={\frac {x-iy}{x^{2}+y^{2}}}} , w=xx2+y2−iyx2+y2{\displaystyle w={\frac {x}{x^{2}+y^{2}}}-i{\frac {y}{x^{2}+y^{2}}}} , Substitute z=x+iy{\displaystyle z=x+iy} into the function. w=ex+iy+e−x−iy{\displaystyle w=e^{x+iy}+e^{-x-iy}} , The expression e−iy{\displaystyle e^{-iy}} is the conjugate of eiy.{\displaystyle e^{iy}.} w=ex(cos⁡y+isin⁡y)+e−x(cos⁡y−isin⁡y){\displaystyle w=e^{x}(\cos y+i\sin y)+e^{-x}(\cos y-i\sin y)} , w=cos⁡y(ex+e−x)+isin⁡y(ex−e−x){\displaystyle w=\cos y(e^{x}+e^{-x})+i\sin y(e^{x}-e^{-x})} , Recall that the hyperbolic functions are defined as cosh⁡x=ex+e−x2{\displaystyle \cosh x={\frac {e^{x}+e^{-x}}{2}}} and sinh⁡x=ex−e−x2.{\displaystyle \sinh x={\frac {e^{x}-e^{-x}}{2}}.} w=2cos⁡ycosh⁡x+i2sin⁡ysinh⁡x{\displaystyle w=2\cos y\cosh x+i2\sin y\sinh x}
2. 2

   Step 2: Substitute z=x+iy{\displaystyle z=x+iy} into the function.

3. 3

   Step 3: Use Euler's formula to decompose the complex exponential.

4. 4

   Step 4: Write w=1/z{\displaystyle w=1/z} in terms of its real and imaginary components.

5. 5

   Step 5: Multiply the numerator and denominator by the complex conjugate and simplify.

6. 6

   Step 6: Separate the real and imaginary components.

7. 7

   Step 7: Write w=ez+e−z{\displaystyle w=e^{z}+e^{-z}} in terms of its real and imaginary components.

8. 8

   Step 8: Use Euler's formula to decompose the complex exponentials.

9. 9

   Step 9: Bring the real and imaginary components together.

10. 10

    Step 10: Simplify using the hyperbolic functions.

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