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Direct computation of the integral of 1/(z-a) around a semicircle and a full circle centred at a

Example

For $r > 0$ and $γ (t) = a + r e^{i t}$ , $\int_{0 \leq t \leq π} \frac{d z}{z - a} = i π, \int_{0 \leq t \leq 2 π} \frac{d z}{z - a} = 2 π i .$ Reversing either orientation negates its value.

Facts & Assumptions

Given: The positively oriented semicircle and circle centred at $a$ .

[L1]

The integer-monomial circle theorem gives the full-circle value $2 π i$ for exponent $- 1$ (On a positively oriented circle about a, the integral of (z-a)^m is zero for every integer m except -1, and is 2 pi i for m=-1).

[L2]

Reversal negates a complex contour integral (Complex line integrals change sign under reversal and add under concatenation).

Verification

technique · direct

1.1

Since $d z = i r e^{i t} d t$ and $z - a = r e^{i t}$ , their quotient is the constant $i$ ; integration over $[0, π]$ and $[0, 2 π]$ gives $i π$ and $2 π i$ .

algebra

2.1

The full-circle value agrees with [L1], and [L2] gives the negative values on reversed paths.

step 1.1L1L2

3.1

The positive-radius hypothesis ensures that the denominator never vanishes.

given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 84 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

A. Weber, Lecture Notes in Complex Analysis, Example 1.7.1 (standard reference, not scraped)