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#REDIRECT[[Calculus of residues]]
{{cleanup}}
'''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.
Alternatively, we may also take note of real and imaginary portions of complex numbers
Cosine is the real portion of a complex number written in cos x + i sin x form
<math>\int e^x \cos x dx =</math>
<math>\int e^x \mathrm{Re}\{ \cos x + i\cdot \sin x \} dx</math>
<math>\int e^x \mathrm{Re}\{ e^{ix} \} dx</math>
<math>\mathrm{Re}\{ \int e^x e^{ix} dx \}</math>
<math>\mathrm{Re}\{ \int e^{(i+1)x} dx \}</math>
{{math-stub}}
[[Category:Integral calculus]]
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