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Exercise:
Determine if there exists an entire function fzuxy+ivxy such that uxyxy+pi. Recall that xyin mathbbR zx+iy and uxy vxy are the real and imaginary parts of f respectively.

Solution:
The Cauchy-Riemann s lead to fracpartial upartial x y fracpartial vpartial y fracpartial upartial y x -fracpartial vpartial x Rightarrow fracpartial vpartial x -x We can then choose vxy fracy^+g_x -fracx^+g_y where g_'x-x and g_'yy. So g_x -fracx^+textconst. g_x fracy^+textconst. This yields to vxy -fracx^+fracy^+c for some cin mathbbR. Finally fz xy+pi-fracix^+iy^+ic c'-fracix^-y^+ixy -fraciz^+c' where c'pi+ic.
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Exercise:
Determine if there exists an entire function fzuxy+ivxy such that uxyxy+pi. Recall that xyin mathbbR zx+iy and uxy vxy are the real and imaginary parts of f respectively.

Solution:
The Cauchy-Riemann s lead to fracpartial upartial x y fracpartial vpartial y fracpartial upartial y x -fracpartial vpartial x Rightarrow fracpartial vpartial x -x We can then choose vxy fracy^+g_x -fracx^+g_y where g_'x-x and g_'yy. So g_x -fracx^+textconst. g_x fracy^+textconst. This yields to vxy -fracx^+fracy^+c for some cin mathbbR. Finally fz xy+pi-fracix^+iy^+ic c'-fracix^-y^+ixy -fraciz^+c' where c'pi+ic.
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ca, cauchy-riemann, entire, eth, holomorphic
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