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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)

The purpose of the Service Type field is to indicate how the datagram is processed Three of the eight bits in this field are used to denote the precedence or level of importance assigned by the sender Thus, this field proves a priority mechanism for the routing of IP datagrams

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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

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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The Identification field enables each datagram or fragmented datagram to be identified If a datagram is fragmented into two or more pieces, the Fragment Offset field specifies the offset in the original datagram of the data being transported Thus, this field indicates where a fragment belongs in the complete message The actual value in this field is an integer that corresponds to a unit of eight octets, providing an offset in 64-bit units

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

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