498 CHAPTER 18. DIFFERENTIAL FORMS

16. Let f : C→ C. Thus if x+ iy ∈ C, f (x+ iy) = u(x,y)+ iv(x,y) where u is the realpart and v is the imaginary part. As in one variable calculus, we define

limh→0

f (z+h)− f (z)h

= f ′ (z)

and we say that this derivative exists exactly when this limit exists. Consider h = itand then let h = t. Take limits in these two ways and conclude that if f ′ (z) exists,then it is given by

f ′ (z) = ux + ivx = vy− iuy

Thus you have the Cauchy Riemann equations ux = vy,vx = −uy. Show that if u,vare both C1, and these Cauchy Riemann equation hold, then the function will bedifferentiable. When this happens, we say the function is analytic. (In fact it can beshown that if the limit of the difference quotient exists, then these real and imaginaryparts will automatically be continuous and the function will be analytic.)

17. For a function f : C→ C which is continuous and γ : [a,b]→ Γ ⊆ C where Γ is apiecewise smooth curve, we define the contour integral∫

Γ

f (z)dz =∫ b

af (γ (t))γ

′ (t)dt.

Show that this equals F (γ (b))−F (γ (a)) if f is analytic, this for some function F .In particular, if Γ is a suitable closed curve, then

∫Γ

f (z)dz = 0. This is Cauchy’stheorem from complex analysis.

18. Suppose f ∈ L1 (U) where U is some open set in Rp. Go ahead and assume f isBorel measurable although it should work with f only Lebesgue measurable. Showthere is a set of mp−1 measure zero N such that if xp ≡ (x1,x2, · · · ,xp−1) /∈ N, thenxp→ f (xp,xp) is in L1

(Uxp

)where Uxp =

{t : (xp, t) ∈U

}.

19. If f ∈ L1 (U) and fxp ∈ L1 (U) where U ⊆ Rp is a box like ∏k (ak,bk). Let fxp referto the weak partial derivative. Can you show that for fixed s, t ∈ (ak,bk) then for a.e.xp, f (xp, t)− f (xp,s) =

∫ ts fxp (xp,τ)dτ? Assume f is Borel measurable.