Splitting an exponential complex number into real and imaginary parts

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SUMMARY

The discussion focuses on splitting the exponential of a complex number, specifically the expression e^{-z^2}, where z is defined as a complex number in the form a + ib. Participants clarify that -z^2 can be expressed as c + id, utilizing Euler's formula, which states e^{c+id} = e^c (cos d + i sin d). This leads to the conclusion that the real part is e^c cos d and the imaginary part is e^c sin d, demonstrating the application of Euler's formula in complex analysis.

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  • Understanding of complex numbers, specifically in the form a + ib.
  • Familiarity with Euler's formula and its applications in complex analysis.
  • Basic knowledge of exponential functions and trigonometric identities.
  • Ability to manipulate complex expressions and convert between forms.
NEXT STEPS
  • Study the derivation and applications of Euler's formula in greater detail.
  • Explore the properties of complex exponentials and their geometric interpretations.
  • Learn about the polar form of complex numbers and its relation to exponential functions.
  • Investigate advanced topics in complex analysis, such as contour integration and residue theorem.
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Mathematicians, physics students, and anyone interested in complex analysis and its applications in various fields, including engineering and signal processing.

dan5
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e-z2

where z is a complex number a+ib
 
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Welcome to PF, dan5! :smile:

Can you calculate -z2?

-z2 should be of the form c + id.

According to Euler's formula, we have ##e^{c+id} = e^c ( \cos d + i \sin d )##.
So the real part is ##e^c \cos d## and the imaginary part is ##e^c \sin d##.
 
Ahhh now I see, thanks to you, and to Euler!
 

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