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20.3 Consequences of Cauchy’s Theorem 713

20.3.7 A Variation on Cauchy’s Integral Formula

We will develop another form of the Cauchy integral formula which we will use in Section 22.4
to write a complex integral formula for the inverse Laplace transform. This will be used to apply
complex analysis to a diffusion problem for a solid cylinder.

THEOREM 20.9 Cauchy Integral Formula—Second Version

Let σ be a real number, and suppose f (z) is differentiable in the half-plane x ≥ σ. Suppose that
there are positive numbers M and n such that

n
|z f (z)|≤ M
for |z| to be sufficiently large (for example, for |z|≥ R for some positive number R).
Then, for any z 0 with Re(z 0 )>σ,
1 σ+ib f (z)
f (z 0 ) =− lim dz.
2π b→∞ σ−ib z − z 0
In this limit, we actually have the integral over the vertical line x =σ, oriented upward (from
−∞ to ∞). This integral of f (z)/(z − z 0 ) over this line is equal to −2πif (z 0 ), hence it is in the
same spirit as the Cauchy integral formula.
We will outline an argument suggesting why this formula is true. Suppose z 0 lies to the right
of the line x = σ. Construct the closed rectangular path C shown in Figure 20.14 having corners
b − ib,b + ib, σ − ib, and σ + ib with b chosen to be sufficiently large, so that C encloses z 0 .
Let C be the path consisting of the upper, lower, and right sides of this rectangle, while L is the
∗
left side of the rectangle, which is the vertical line from σ −ib to σ +ib. By the Cauchy integral
formula,
1 f (z)
f (z 0 ) = dz
2πi C z − z 0
1 f (z) σ+ib f (z)
= dz − dz .
2πi C ∗ z − z 0 σ−ib z − z 0

y
σ + ib
b + ib

L
z 0 x = b
x = σ
x

b – ib
σ – ib

FIGURE 20.14 C and C in this Cauchy
∗
integral representation.

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October 14, 2010 15:32 THM/NEIL Page-713 27410_20_ch20_p695-714