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  "body": "\n## Example\n\nLet $\\omega$ be a real number.\nThe [[Laplace transforms|05XT5AVJBK]] of [[sine and cosine|0WB57D9488]] are\n$$\\mathcal{L}\\{\\cos \\omega t\\}(s) = \\frac{s}{s^2 + \\omega^2}, \\qquad \\mathcal{L}\\{\\sin \\omega t\\}(s) = \\frac{\\omega}{s^2 + \\omega^2} \\quad (\\operatorname{Re} s \u003e 0).$$\nCheck: write $\\cos \\omega t = \\frac{1}{2}(e^{i\\omega t} + e^{-i\\omega t})$ and $\\sin \\omega t = \\frac{1}{2i}(e^{i\\omega t} - e^{-i\\omega t})$.\nBy the [[transform of the exponential|X5JYH26J3M]], $\\mathcal{L}\\{e^{\\pm i\\omega t}\\}(s) = 1/(s \\mp i\\omega)$ for $\\operatorname{Re} s \u003e 0$.\nBy [[linearity|SHX9JPKJBE]],\n$$\\mathcal{L}\\{\\cos \\omega t\\}(s) = \\frac{1}{2}\\left(\\frac{1}{s - i\\omega} + \\frac{1}{s + i\\omega}\\right) = \\frac{s}{s^2 + \\omega^2},$$\n$$\\mathcal{L}\\{\\sin \\omega t\\}(s) = \\frac{1}{2i}\\left(\\frac{1}{s - i\\omega} - \\frac{1}{s + i\\omega}\\right) = \\frac{\\omega}{s^2 + \\omega^2}.$$\nFor $\\omega \\ne 0$ both transforms have poles at $s = \\pm i\\omega$ on the imaginary axis, which matches the abscissa of convergence $0$ of these bounded, non-decaying functions.\n\n## Explanations\n\n### L1 human\n\nOscillations of angular frequency $\\omega$ show up on the transform side as a pair of poles at $\\pm i\\omega$.\nThe cosine starts at $1$ and the sine starts at $0$, which is why the cosine has $s$ in the numerator and the sine has the constant $\\omega$.\nThese two lines of the table are what makes the Laplace transform useful for springs and electric circuits.\n",
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