- Verify all the usual rules of differentiation including the product and chain
- Suppose f and f′ : U → ℂ are analytic and f =
Verify uxx + uyy = 0 and vxx + vyy = 0. This partial differential equation
satisfied by the real and imaginary parts of an analytic function is called
Laplace’s equation. We say these functions satisfying Laplace’s equation are
harmonic functions. If u is a harmonic function defined on B show that
dt is such that u + iv is analytic.
- Let f : U → ℂ be analytic and f =
. Show u,v and uv
are all harmonic although it can happen that u2 is not. Recall that a function,
w is harmonic if wxx + wyy = 0.
- Define a function f
≡z ≡ x − iy where z = x + iy. Is f analytic?
- If f =
f is analytic, verify that
- Show that if u +
f is analytic, then
∇u ⋅∇v = 0. Recall
- Show that every polynomial is analytic.
- If γ =
iy is a
C1 curve having values in U, an open set of ℂ, and if
f : U → ℂ is analytic, we can consider f ∘γ, another C1 curve having values in ℂ.
Also, γ′ and
′ are complex numbers so these can be considered
as vectors in
ℝ2 as follows. The complex number, x + iy corresponds to
the vector, 〈x,y〉. Suppose that γ and η are two such C1 curves having
values in U and that γ =
z and suppose that f : U → ℂ is
analytic. Show that the angle between
′ is the
same as the angle between
η′ assuming that
Thus analytic mappings preserve angles at points where the derivative is
nonzero. Such mappings are called isogonal. . Hint: To make this easy to
show, first observe that 〈x,y〉⋅〈a,b〉 =
z = x + iy and
w = a + ib.
- Analytic functions are even better than what is described in Problem 8. In addition
to preserving angles, they also preserve orientation. To verify this show that if
z = x + iy and w = a + ib are two complex numbers, then 〈x,y,0〉 and 〈a,b,0〉 are
two vectors in ℝ3. Recall that the cross product, 〈x,y,0〉×〈a,b,0〉, yields a vector
normal to the two given vectors such that the triple, 〈x,y,0〉,〈a,b,0〉, and
〈x,y,0〉×〈a,b,0〉 satisfies the right hand rule and has magnitude equal to the
product of the sine of the included angle times the product of the two
norms of the vectors. In this case, the cross product either points in the
direction of the positive z axis or in the direction of the negative z axis.
Thus, either the vectors 〈x,y,0〉,〈a,b,0〉,k form a right handed system or
the vectors 〈a,b,0〉,〈x,y,0〉,k form a right handed system. These are
the two possible orientations. Show that in the situation of Problem 8
the orientation of γ′
,k is the same as the orientation of the
,k. Such mappings are called conformal.
If f is analytic and f′
≠0, then we know from this problem and the
above that f is a conformal map. Hint: You can do this by verifying that
×η′. To make the verification
easier, you might first establish the following simple formula for the cross product
x + iy = z and a + ib = w.
- Write the Cauchy Riemann equations in terms of polar coordinates. Recall the
polar coordinates are given by
This means, letting u =
, write the Cauchy Riemann
equations in terms of r and θ. You should eventually show the Cauchy Riemann
equations are equivalent to
- Show that a real valued analytic function must be constant.