Integration using Euler's formula: Difference between revisions

Revision as of 14:29, 28 August 2008

Revision as of 15:14, 30 August 2008

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'''Integration using complex analysis''' is a method of integrating certain functions.

Suppose we wanted to integrate:

: <math>\int e^x \cos x \, dx</math>

Instead of using [[Integration by parts]], we may substitute the cosine function for its Euler form: <math>\cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2}</math>

: <math>\int e^x \cdot \frac{e^{ix} + e^{-ix}}{2} \, dx</math>

: <math>{1\over 2} \int e^{x(1+i)} + e^{x(1-i)} \, dx</math>

This is far easier to integrate.