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If for every real number E there exists a real number r such that If ( z ) - fol < E for all lz( > r, then we say that sm,+, f ( z ) = fo. Let This function is undefined at z = 0. Show that limz-,o f ( z ) fails to exist. Sobtion. have 59 Let us move toward the origin along the y-axis. With x = 0 in f ( z ) , we f(z> = + i(y2 Y ) = i(y Y + 1). What the definition asserts is that the magnitude of the difference between f ( z ) and fo can be made smaller than any preassigned positive number E provided the point representing z lies more than the distance r from the origin.

Contrast why one does not say that lim,,o this to the result in (a). d) The definition used above and in parts (a) and (b) cannot be used for functions whose limits at infinity are infinite. Here we modify the definition as follows: We say that lirn,,, f ( z ) = oo if, given p > 0, there exists r > 0 such that I f(z)l > p for all r < lz I . In other words, one can make the magnitude of f ( z ) exceed any preassigned positive real number p if one is at any point at least a distance r from the origin.

Assuming that ei is differentiable, we follow the chain rule and get @ dz - dez d i The first term on the right is ei, while the second does not exist. zl-z. ) Thus ei is not differentiable and is nowhere analytic. 4-1 ---,. v ------m-=-- + times instead of expressing a function of z in the form f ( z ) = u ( x , Y) Y), it is convenient to change to the polar system r, 0 so that x = r cos 0, 76 Chapter 2 ~h~ complex Function and Its Derivative + y = r sin 0. ~ h u fs( z ) = u(r, 0 ) iv(r, 0 ) .