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1) it should have highest amplitude on y axis at x=0. It should be symmetrical about y axis. It should fade away at some x on both positive and negetive x axis symmetrically. Fading in a way of decreasing amplitude. It should not give negetive value of y.

2) most important. The square of norm of μ should be normalized in infinity.

I am not good at math but this is integral part of my project. It can be visualized as : consider x axis as still surface of water and you drop a pebble on it. But it should be shifted upwards such that it should not fall in negetive y axis. Also it should be preferrably in cartesian co ordinates