11.7. EXERCISES 271

It is a continuous function of w and equals 1 at 0 = φ (z0) so, since it is integer valued, itequals 1 on all of B(0,ε) , but this is the number of zeroes of φ (z)−w. Thus φ (B(z0,δ )) =B(0,ε). Hence, φ

m (B(z0,δ )) = B(0,εm). It follows that

f (B(z0,δ )) = f (z0)+B(0,εm) = B( f (z0) ,εm)

and so this shows that f maps small open balls to open balls. Thus f (Ω) is a connectedopen set.

11.7 Exercises1. A fractional linear transformation is one of the form f (z) = az+b

cz+d where ad−bc ̸= 0where a,b,c,d are in a field, say R,Q,C. Let M denote the 2× 2 invertible ma-trices having entries in the same field. Denote by F these fractional linear trans-

formations. For A =

(a bc d

)∈ M, let φ (A)(z) ≡ az+b

cz+d . Show that φ (AB)(z) =

φ (A)◦φ (B)(z). Show that φ (I)(z) = z and that φ : M→ F is onto. Show φ (A)−1 =φ(A−1

)so there is an easy way to invert such a fractional linear transformation. This

problem is best for those who have had a beginning course in linear algebra.

2. The modular group consists of functions f (z) = az+bcz+d where a,b,c,d are integers and

ad−bc = 1. Surprisingly, each of these has an inverse. f−1 (z) = dz−b−cz+a . Verify that

this is the case. This means f−1 ◦ f (z) = z. Show also that if f ,g are two of these,then f ◦ g is another one. This last part might be a little tedious without the aboveproblem.

3. Suppose U is an open set in C and f : U → C is analytic, ( f ′ (z) exists for z ∈U).For z = x + iy, f (z) = f (x+ iy) = u(x,y) + iv(x,y), u, and v having real values.These are the real and imaginary parts of f . The partial derivative, ux is defined byfixing y and considering only the variable x. ux (x,y) ≡ limh→0

u(x+h,y)−u(x,y)h . Other

versions of this notation are similar. Thus partial derivatives are a one variable con-sideration. Show that the existence of f ′ (z) implies the Cauchy Riemann equations.ux = vy,uy =−vx. Hint: In the difference quotient for finding f ′ (z) , use h→ 0 andthen ih→ 0 for h real.

4. Suppose t→ z(t) = x(t)+ iy(t) and t→ w(t) = x̂(t)+ iŷ(t) are two smooth curveswhich intersect when t = 0. Then consider the two curves t → f (z(t)) and t →f (w(t)) where f is analytic near z(0) = w(0) . Show the cosine of the angle betweenthe resulting two curves is the same as the cosine of the angle of the original twocurves when t = 0. Hint: You should write f (z) = u(x,y) + iv(x,y) and use theCauchy Riemann equations and the chain rule. This problem is really dependent onknowing a little bit about functions of more than one variable so does not exactly fitin this book. It depends on remembering some elementary multivariable calculus.Also recall that the cosine of the angle between two vectors u,v is (u ·v)/ |u| |v|.That analytic mappings preserve angles is very important to some people.

5. Suppose f : C→ C has a derivative at every point and f ′ (z) = 0 for all z. Showthat, as in the case of a real variable, f (z) is a constant. Generalize to an arbitraryopen connected set. Hint. Pick z0 and for arbitrary z, consider z0 + t (z− z0) where