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UseaFouriertransformtosolvetheLaplaceequationontheupper-half plane:

?^2u= ?u= 0

u(x,0) =f(x).

Your solution must be written as a convolution of the function f(x) and some kernel.

Hint: The Fourier transform ODE will yield a sum of two familiar solutions.Notice that this solution must be bounded.Therefore, you can cleverly combine these solutions into one of form C(?)h(|?|), where his a function you'll recognize.The following Fourier transform will then help you:

Question 14. Use a Fourier transform to solve the Laplace equation on the upper-half plane: V211, = Au = 0 \"(06,0) = f($)- Your solution must be written as a convolution of the function f (m) and some kernel. Hint: The Fourier transform ODE will yield a sum of two familiar solutions. Notice that this solution must be bounded. Therefore, you can cleverly combine these solutions into one of form C(w)h(|w|), where h is a flmction you'll recognize. The following Fourier transform will then help you: 2 a _ |w| 1\" [Jim] 'exp ['7], for a > 0