Question 9: (a) (z-5i)^2 is a polynomial, hence continuous everywhere, and thus on e can simply substitute in the limit to obtain ((2+3i)^2 - 5i)^2 = -8i.
(c) This is an indeterminate form; one can either use L'hopital's rule (which is valid for complex functions, although we will not prove this until much later), or factorize the numerator as z^2 + 9 = (z-3i)(z+3i) and simplify the limit to z+3i. Now the function is continuous and we can simply substitute to obtain 6i.
(e) The function in the limit can be simplified to 2z_0 + /\z, which (being a polynomial in /\z) converges to 2z_0 as /\z tends to zero. Alternatively, one can note that this is precisely the derivative of z^2 at z_0, which is equal to 2z_0 since the differentiation of polynomials proceeds in complex analysis in exactly the same manner as in real analysis.
Question 15: The limit along the real approach y=0 is 2i, and the limit along the imaginary approach x=0 is 2i+1, which are not equal, so the full limit does not exist.
Question 4: (a) The expression in the limit of Definition 4 simplifies to /\x / /\ z, where /\z = /\x + i /\y is the Cartesian form of /\z. Taking the limits along the real and imaginary approaches shows that this limit does not exis t.
(b) is proven in the same way as (a).
(c) is trickier, and one needs to divide into two cases, z_0 = 0 and z_0 != 0. Wh en z_0 = 0 the argument is similar to (a) and (b). When z_0 is nonzero one needs a t rick to simplify the expression, namely to multiply the numerator and denominator by |z_0 - /\z| + |z_0|. One can use the identity |z|^2 = zz* to simplify the numerat or, while the |z_0 - /\z| + |z_0| term in the denominator converges to 2|z_0| and can be taken out of the limit. The remaining terms in the expression will eventually simplify to 2z* + /\z + /\z* / /\z, which does not converge because the first two terms converge and the last term diverges. (The sum of a convergent limit and a divergent limit is always divergent).
Question 2: Write h = u+iv. Since the partial derivatives of u and v are polynomials, they are continuous, and so differentiability is equivalent to the Cauchy-Riemann equations being satisfied. In this case these equations become
3x^2 + 3y^2 - 3 = 3y^2 + 3x^2 - 3
6xy = -6xy.The first equation is always satisfied, and the second one is satisfied exactly when x = 0 or y = 0. Thus h is differentiable on the co-ordinate axes and nowhere else. To be analytic at a point, you have to be differentiable on a ball centered at that point; however, the co-ordinate axes do not contain any balls, and so there is no place where the function can be analytic.
Question 5: The partial derivatives, being products of polynomials,
exponential functions, sines and cosines, are continuous, and so it suffices
to verify that the Cauchy-Riemann equations hold. Once one does
this, one can use the identity df/dz = df/dx = 1/i df/dy to find the
derivative. Alternatively, one can observe that f(z) can be written
much more compactly as exp(z^2), so f is evidently differentiable everywhere
with a derivative of 2z exp(z^2).
Question 11: Write f = u+iv. Since f is analytic in D, f is differentiable in D and so therefore f obeys the Cauchy-Riemann equations du/dx = dv/dy, du/dy = -dv/dx. Similarly f* = u-iv must also satisfy the Cauchy-Riemann equations, so du/dx = -dv/dy and du/dy = dv/dx. Combining these equations together and simplifying we see that both partial derivatives of u and v are zero, which (since D is a domain) implies that u and v are both constant, so that f is constant.
Question 1: (a) f = u+iv where u(x+iy) = x^2 - y^2 + 2x + 1 and v(x+iy) = 2xy + 2y; du^2/dx^2 + du^2/dy^2 = 2 - 2 = 0 and dv^2/dx^2 + dv^2/dy^2 = 0 + 0 = 0.
(c) h = u+iv where u(x+iy) = exp(x) cos y and v(x+iy) = exp(x) sin y;
du^2/dx^2 + du^2/dy^2 = exp(x) cos y - exp(x) cos y = 0 and
dv^2/dx^2 + dv^2/dy^2 = exp(x) sin y - exp(x) sin y = 0/
Question 3: (c) u is the imaginary part of exp(z), so it is the real part of -iexp(z). Since -i exp(z) is evidently analytic, u is automatically harmonic, and the imaginary part of -i exp(z), -exp(x) cos(y), is a conjugate of u.
(e) u is the real part of Log(z), which is analytic when Re z > 0, so u is automatically harmonic, and the imaginary part of Log(z), which is arctan(y/x) when Re z > 0, is a conjugate of u. (You can also get this result by directly solving the Cauchy-Riemann equations, but you have to be careful of the fact that y could be zero, so one must look out for any expressions with y in the denominator.)
Question 12: There are several ways to do this. One approach is as follows:
w = z / (sqrt(2) - exp(pi i/4) z);other solutions are possible.