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Complex Variables Demysti ed
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real part of z and y the imaginary part of z, but both x and y are themselves real variables This concept carries over to a complex function, which can be written in the form f ( z ) = f ( x + iy) = u( x , y) + iv ( x , y) The real part of f is given by Re( f ) = u( x , y) And the imaginary part of f is given by Im( f ) = v ( x , y) (24) (23)
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Notice that we can write down the complex conjugate of a function With f ( z ) = f ( x + iy) = u( x , y) + iv ( x , y) the complex conjugate is given by f ( z ) = f ( x + iy) = u( x , y) iv ( x , y) (25)
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The same rule applied to complex numbers and complex variables was used, namely, we let i i in order to obtain the complex conjugate Note that u( x , y) and v ( x , y) are unchanged by this operation because they are both real functions of the real variables x and y In chap 1, we learned how to write the real and imaginary parts of z in terms of z , z using Eqs (111) and (112) We can write down analogous formulas for the real and imaginary parts of a function First let s consider the real part of a complex function We can add the function to its complex conjugate f + f = u + iv + u iv = 2u Hence the real part of a complex function is given by u ( x , y) = f (z) + f (z) 2 (26)
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And we can take the difference between a function and its complex conjugate: f f = u + iv (u iv ) = 2iv This gives us the imaginary part of a complex function: v ( x , y) = f (z) f (z) 2i (27)
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EXAMPLE 24 What is the complex conjugate of f(z) = 1/z
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SOLUTION First of all we can write the function as f ( z ) = f ( x + iy) = Therefore the complex conjugate is 1 1 1 f = = x iy = z x + iy EXAMPLE 25 What are the real and imaginary parts of f ( z ) = z + (1/z ) SOLUTION We let z = x + iy Then we have f ( x + iy) = x + iy + 1 x + iy 1 x + iy
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We need to write the second term in standard Cartesian notation This is done by multiplying and dividing by its complex conjugate: 1 1 x iy x iy = x iy = x 2 + y 2 x + iy x + iy So, we have f ( x + iy ) = x + iy + =x+ =
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x iy x 2 + y2
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iy x + iy 2 2 x + y2 x +y
x 3 + xy 2 + x y3 + x 2 y y + i x 2 + y2 x 2 + y2
So the real part of the function is Re( f ) = x 3 + xy 2 + x = u ( x , y) x 2 + y2
Complex Variables Demysti ed
The imaginary part of the function is Im( f ) = y3 + x 2 y y = v ( x , y) x 2 + y2
Note that we can write the real and imaginary parts in terms of z , z as follows We have f (z) = z + 1 1 =z+ z z
Now 1 1 z z f + f =z+ +z + =z+z + + zz zz z z z+z z+z = zz + zz zz = So Re( f ) = f + f z 2 z + zz 2 + z + z = 2 2 zz z 2 z + zz 2 + z + z zz
To get the imaginary part we calculate 1 1 z z f f = z + z + = z + z zz zz z z z+z z+z = zz zz zz = z 2 z + zz 2 z z zz