I think you will need to apply Euler's Forumla here...
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Here .
I want to integrate the following function: e^(-i*(u*x+v*y)) with respect to x and y. Evaluated where x value is between -W and W, while y value is between -H and H. And i in the function is the imaginary number, while u and v are arbitrary constant variable.
I was able to get the integral itself, which evaluates to: -e ^(-i*(u*x + v*y))/(u*v)
However mathematica was able to simplify the final solution to: (4 Sin[H v] Sin[u W])/(u v)
Whereas the furthest I was able to get was: 4 * e ^ ( i * (u*w - v *h))/(uv)
Where did I go wrong? What am I missing?
Any help is greatly appreciated.
Thank you!
Beautiful, that's probably the key.
I'll get on it, thanks!
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Sorry that's still a no go... Please take a look at the scribbles above. Even with the Euler's formula I wasn't able to drop the terms sufficiently to leave only the Sine functions like the Mathematica's equation.
Equation: Integrate[E^(-I *(u*x + v * y)), {x, -W, W}, {y, -H, H}]
Result: (4 Sin[H v] Sin[u W])/(u v)
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Finally was able to get the right result, as it turned out I'm forgetting my trig identities, mainly:
Cos(-x) = Cos (x)
also
Cos(x+y) = Cos(x)Cos(y) - Sin(x) Sin(y)
>.< doh! In both instances of the identities I was confusing the plus (+) for the minus (-) sign.
At any rate here's the corrected simplification:
p.s. I should have carried over the (uv) denominator, but I just put that back at the final step.
Thank you all!