MMF 2018

Tarea V en Métodos Matemáticos para la Fı́sica y la Astronomı́a
June 4, 2019
Fecha: 19 November 2018
Let R be a simply connected region of R2 with Cartesian coordinates (x, y). Define complex
coordinates (z, z̄) in R by z := x + iy and z̄ := x − iy. Let f : R → C be smooth. Its complex
partial derivatives are defined by
f (z, z̄ + ¯) − f (z, z̄)
,
→0
¯
f (z + , z̄) − f (z, z̄)
,
→0
∂z̄ f := lim
∂z f := lim
where = || exp(iArg()) and the limits amount to taking || → 0 at fixed Arg(); the smoothness
of f implies that its complex derivatives are well-defined, i.e. independent of Arg().
Problem 1.
a) Show that
1
∂z f = (∂x − i∂y )f ,
2
1
∂z̄ f = (∂x + i∂y )f .
2
b) Show that f is holomorphic, i.e.
∂z̄ f = 0 ,
iff its real and imaginary parts obey the Cauchy–Riemann equations, viz.
∂x u − ∂y v = 0 = ∂y u + ∂x v ,
f (x, y) ≡ u(x, y) + iv(x, y) ,
u, v ∈ R .
c) Show that a monomial in (x, y) of degree n that is holomorphic is of the form az n where a is a
complex constant.
d) Show that if n ∈ Z and a is a complex constant, then az n is holomorphic.
Problem 2.
a) Show that if f is holomorphic, then its (total) differential
df := (dx∂x + dy∂y )f = dz∂z f ,
dz := dx + idy .
b) Show that if i) Ca→b ⊂ R is a path from a to b with parametrization γ : [0, 1] → Ca→b such that
z(γ(0)) = a ,
z(γ(1)) = b ,
∂t z(γ(t)) 6= 0 ;
and ii) f has a primitive function F , viz. f (z) = ∂z F (z), then
dzf (z)|Ca→b = dt∂t F (z(γ(t))) .
c) Use (b) to show that
Z
dzf (z) = F (b) − F (a) .
Ca→b
1
Problem 3. a) Show that if i) Σ ⊂ R is a square in R with sides parallel to the x and y axes; and
ii)
H ω := dxωx (x, y) + dyωy (x, y) is a smooth line element on R, then the closed contour integral
ω of ω along the boundary of Σ obeys
∂Σ
I
ZZ
ω=
dω ,
dω := dx ∧ dy(∂x ωy − ∂y ωx ) ,
∂Σ
Σ
where dx ∧ dy is the area form on Σ.
e is an invertible smooth map from R to a region R
e ∈ R2 with
b) Assume that i) ϕ : R → R
e respectively,
Cartesian coordinates (x̃, ỹ); and ii) ω and ω
e are smooth line elements on R and R,
that are related by
ω|p = ω
e |ϕ(p) , for all p ∈ R .
Show that the components of ω and ω
e are related by the transformation rule
ωx (x, y)
∂x x̃ ∂x ỹ
ω
ex̃ (x̃, ỹ)
=
,
ωy (x, y)
∂y x̃ ∂y ỹ
ω
eỹ (x̃, ỹ)
where the Jacobian is defined by expressing (x̃, ỹ) as functions of (x, y) via the map ϕ, viz.
x̃(ϕ(p)) = x̃(x(p), y(p)) idem ỹ.
c) Show that the above transformation rule implies that
dω|p = de
ω |ϕ(p) ,
for all p ∈ R ,
where de
ω := dx̃ ∧ dỹ(∂x̃ ω
eỹ − ∂ỹ ω
ex̃ ) and the area form transforms as follows:
∂x x̃ ∂x ỹ
(dx̃ ∧ dỹ)|ϕ(p) = det
(dx ∧ dy)|p .
∂y x̃ ∂y ỹ
d) Show that if C is an oriented curve in R with image ϕ(C) in R̃, then
Z
Z
ω=
ω
e.
C
ϕ(C)
e) Show that if Σ ⊂ R is a simply connected region in R with image ϕ(Σ) in R̃, then
ZZ
ZZ
dω =
de
ω.
Σ
ϕ(Σ)
f) Argue that ϕ can be choosen such that ϕ(Σ) is a square; combine this with (a) to show that
ZZ
ZZ
I
I
dω ,
ω=
ω
e=
de
ω=
∂Σ
that is
ϕ(Σ)
ϕ(∂Σ)
I
Σ
ZZ
ω=
dω ,
∂Σ
Σ
known as Stokes’ theorem.
g) Use Stokes’ theorem to show that if C is a closed contour in R, then
I
dzf = 0 ,
C
known as Cauchy’s integral theorem.
h) Use Cauchy’s integral theorem to show that
Z
F (z) :=
dsf (s)
Cz0 →z
is a (single-valued) primitive function for f in R.
2