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Im

Denotes the imaginary part of a complex number in mathematical notation.

Overview

Essential in complex analysis and related mathematical fields where complex numbers are manipulated and analyzed.

  • Commonly paired with its counterpart Re for expressing complex number components
  • Frequently appears in signal processing, electrical engineering, and quantum mechanics
  • Used in theoretical mathematics to extract the imaginary component from expressions
  • Often seen in equations involving wave functions and Fourier transforms

Examples

Expressing the imaginary part of a complex number z.

ℑ(z)=b where z=a+bi\Im(z) = b \text{ where } z = a + bi 
\Im(z) = b \text{ where } z = a + bi

Finding the imaginary component of a complex exponential.

ℑ(eiθ)=sin⁡(θ)\Im(e^{i\theta}) = \sin(\theta) 
\Im(e^{i\theta}) = \sin(\theta)

Showing the imaginary part in quantum mechanics wave function.

ℑ(ψ(x,t))=ℏ2mωsin⁡(ωt)\Im(\psi(x,t)) = \sqrt{\frac{\hbar}{2m\omega}} \sin(\omega t) 
\Im(\psi(x,t)) = \sqrt{\frac{\hbar}{2m\omega}} \sin(\omega t)