Complex Integration by Substitution
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Theorem
Let $\closedint a b$ be a closed real interval.
Let $\phi: \closedint a b \to \R$ be a real function which has a derivative on $\closedint a b$.
Let $f: A \to \C$ be a continuous complex function, where $A$ is a subset of the image of $\phi$.
If $\map \phi a \le \map \phi b$, then:
- $\ds \int_{\map \phi a}^{\map \phi b} \map f t \rd t = \int_a^b \map f {\map \phi u} \map {\phi'} u \rd u$
If $\map \phi a > \map \phi b$, then:
- $\ds \int_{\map \phi b}^{\map \phi a} \map f t \rd t = -\int_a^b \map f {\map \phi u} \map {\phi'} u \rd u$
Proof
Let $\Re$ and $\Im$ denote real parts and imaginary parts respectively.
Let $\map \phi a \le \map \phi b$.
Then:
\(\ds \int_{\map \phi a}^{\map \phi b} \map f t \rd t\) | \(=\) | \(\ds \int_{\map \phi a}^{\map \phi b} \map \Re {\map f t} \rd t + i \int_{\map \phi a}^{\map \phi b} \map \Im {\map f t} \rd t\) | Definition of Complex Riemann Integral | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_a^b \map \Re {\map f {\map \phi u} } \map {\phi'} u \rd u + i \int_a^b \map \Im {\map f {\map \phi u} } \map {\phi'} u \rd u\) | Integration by Substitution/Definite Integral | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_a^b \map \Re {\map f {\map \phi u} \map {\phi'} u} \rd u + i \int_a^b \map \Im {\map f {\map \phi u} \map {\phi'} u} \rd u\) | Multiplication of Real and Imaginary Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_a^b \map f {\map \phi u} \map {\phi'} u \rd u\) |
Let $\map \phi a > \map \phi b$.
Then:
\(\ds \int_{\map \phi b}^{\map \phi a} \map f t \rd t\) | \(=\) | \(\ds \int_{\map \phi b}^{\map \phi a} \map \Re {\map f t} \rd t + i \int_{\map \phi b}^{\map \phi a} \map \Im {\map f t} \rd t\) | Definition of Complex Riemann Integral | |||||||||||
\(\ds \) | \(=\) | \(\ds -\int_a^b \map \Re {\map f {\map \phi u} } \map {\phi'} u \rd u - i \int_a^b \map \Im {\map f {\map \phi u} } \map {\phi'} u \rd u\) | Integration by Substitution/Definite Integral | |||||||||||
\(\ds \) | \(=\) | \(\ds - \paren{ \int_a^b \map \Re {\map f {\map \phi u} \map {\phi'} u} \rd u + i \int_a^b \map \Im {\map f {\map \phi u} \map {\phi'} u} \rd u }\) | Multiplication of Real and Imaginary Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds -\int_a^b \map f {\map \phi u} \map {\phi'} u \rd u\) |
$\blacksquare$
Sources
- 2001: Christian Berg: Kompleks funktionsteori $\S 2.2$