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I want to calculate some convolution like integrations:

[itex]g_1(k,l)=\int\int_A \cos(k(x+y))\cos(ly) f(x,y)dx dy[/itex]

[itex]g_2(k,l)=\int\int_A \cos(k(x+y))\sin(ly) f(x,y)dx dy[/itex]

[itex]g_3(k,l)=\int\int_A \sin(k(x+y))\cos(ly) f(x,y)dx dy[/itex]

[itex]g_4(k,l)=\int\int_A \sin(k(x+y))\sin(ly) f(x,y)dx dy[/itex]

[itex] f(x,y) =\cos(k(x-x_0)) \cos(l(y-y_0)) - b [/itex]

[itex] |x-x_0|< \pi/k, |y-y_0|< \pi/l [/itex]

[itex]k,l[/itex] are integers, and [itex]x_0,y_0,b [/itex] are constant real numbers, [itex]0<b<1[/itex]. Region A is the area where [itex]f(x,y)\geq 0[/itex].

I thought about transforming the coordinate into curvilinear coordinate, so that the two base vectors are tangent and normal to the line [itex] f(x,y) [/itex]. But I don't know how to derive.

Can anyone provide some help and guidance? Thanks!

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# Integration of function in a region

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