2. If f(t)is continuous and ta< tc< tb, then
∫tb
ta
f(t)dt =
∫tc
ta
f(t)dt +
∫tb
tc
f(t)dt
3. The modulus of the integral is less than or equal to the integral of the modulus
∣∣
∣∣
∫tb
ta
f(t)dt
∣∣
∣∣≤
∫tb
ta
|f(t)|dt
4. If F=F(t)is such that dF
dt
=F′(t) = f(t)for ta≤t≤tb, then
∫tb
ta
f(t)dt =F(t)]ttab=F(tb)−F(ta)
5. If G(t)is defined G(t) =
∫t
ta
f(t)dt, then dG
dt
=G′(t) = f(t)
6. The conjugate of the integral is equal to the integral of the conjugate
∫ tb
ta
f(t)dt =
∫tb
ta
f(t)dt
7. Let f(t, τ )denote a function of the two variables tand τwhich is defined and con-
tinuous everywhere over the rectangular region R={(t, τ )|ta≤t≤tb, τc≤τ≤τd}.
If g(τ) =
∫tb
ta
f(t, τ )dt and the partial derivatives of fexist and are continuous on
R, then
dg
dτ
=
∫tb
ta
∂f (t, τ )
∂τ
dt
which shows that differentiation under the integral sign is permissible.
Assume F(z) is an analytic function with derivative f(z) = dFdz and z=z(t) for
t 1 ≤t≤t 2 is a piecewise smooth arc C in a region Rof the z-plane, then one can
write ∫
C
f(z)dz =
∫t 1
t 1
dF
dz dz =F(z(t))
t 2
t 1
=F(z(t 2 )) −F(z(t 1 ))
This is a fundamental integration property in the z-plane. Note that if F(z) = U+iV
is analytic and f(z) = u+iv , then one can write
F′(z) = ∂U
∂x
+i∂V
∂x
=u+iv =∂V
∂y
−i∂U
∂y
