Question about the argument in a Complex Exponential

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SUMMARY

The discussion centers on the application of Euler's formula, specifically the expression e^{-iwx} and its equivalence to cos(-wx) - isin(wx). Participants confirm that by substituting u = wx, the relationship holds true as e^{-iu} = cos(u) - i*sin(u). Additionally, it is emphasized that since cosine is an even function, the negative sign can be eliminated from the argument. The discussion clarifies that Euler's formula is applicable regardless of whether x is a real or complex number.

PREREQUISITES
  • Understanding of Euler's formula: e^{ix} = cos(x) + i*sin(x)
  • Basic knowledge of complex numbers and their properties
  • Familiarity with trigonometric functions, particularly the even nature of cosine
  • Experience with substitution methods in calculus
NEXT STEPS
  • Explore advanced applications of Euler's formula in complex analysis
  • Study the properties of even and odd functions in trigonometry
  • Learn about the implications of complex exponentials in signal processing
  • Investigate the use of substitution techniques in solving differential equations
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Mathematicians, physics students, and anyone interested in complex analysis or trigonometric identities will benefit from this discussion.

Haku
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Homework Statement
e^-iwt = ?
Relevant Equations
Eulers formula
I know that e^-ix = cos(-x)-isin(x), but if we have e^-iwx does that equal cos(-wx) - isin(wx)?
Thanks
 
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Sure, why not? Just do a u-sub. Let ##u = wx## then you have ##e^{-iu} = \cos u - i\sin(u)##. Also, remember that cos is an even function, so you can drop the negative inside your argument!
 
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If w is supposed to be a fixed complex number, you might prefer to rewrite the expression a bit before applying Euler's formula to it. But the statement is certainly true, nothing about Euler's formula requires x to be a real number to begin with.
 
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