where it has been assumed that <math>g</math> can be broken downdecomposed into functionsthe analyticsum inof thetwo lower-half planefunctions: <math>g_{+}</math> andwhich is analytic in the upper-half half-plane, and <math>g_{-}</math>, respectivelywhich is in the lower half-plane. Now, as the left-hand side of the above equation is analytic in the lower-half plane, whilst the right-hand side is analytic in the upper-half plane, analytic continution guarantees existence of an entire function which coincides with the left- or right-hand sides in their respective half-planes. Furthermore, since it can be shown that the functions on either side of the above equation decay at large <math>k</math>, an application of [[Liouville's theorem (complex analysis)|Liouville's theorem]] shows that this entire function is identically zero, therefore
plane, whilst the right-hand side is analytic in the upper-half plane, analytic continution guarantees existence of an entire function which coincides with the left- or right-hand sides in their respective half-planes. Furthermore, since it can be shown that the functions on either side of the above equation decay at large <math>k</math>, an application of [[Liouville's theorem (complex analysis)|Liouville's theorem]] shows that this entire function is identically zero, therefore
