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so if you have a wave that is e to the J K X okay so you X T X 0 is equal to that then instead of a plugging in so lets lets first the plugging into the analytical derivative du u DX is equal to j k times u right but if you plug it into for example a central difference scheme okay that is equal to something different so you I plus 1 would be equal to e to the JK x + JK Delta X and U of I minus 1 would be equal to j KX minus JK Delta X right because I plus 1 and I minus 1 are spaced from X by Delta X and minus Delta X divided by Delta X and then this term and this sorry this is e and this term can be pulled out because you can convert the exponential of summation and and into the product of Exponentials so it is equal to e to the JK x times e to the JK Delta X minus e to the J K minus Delta X right ok so this is actually just a U at X its equal to u times and this is you can work out the real part of this actually gets cancelled out the imaginary part of this is actually J times sin