HeavisideTheta 积分表示

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HeavisideTheta integral representation
$$\theta(x)=\frac{\lim _{\varepsilon \rightarrow 0} \frac{1}{\varepsilon} \int_{-x}^{\infty} e^{-t^{2} / \varepsilon^{2}} d t}{\sqrt{\pi}}$$

1. HeavisideTheta[x] == Limit[Integrate[E^(-(t^2/ε^2)), {t, -x, Infinity}]/ε, ε -> 0]/Sqrt[Pi]

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$$\theta(x)=\frac{\lim _{\varepsilon \rightarrow 0} \int_{-\infty}^{x} \frac{\sin \left(\frac{t}{\varepsilon}\right)}{t} d t}{\pi}$$

1. HeavisideTheta[x] == Limit[Integrate[Sin[t/ε]/t, {t, -Infinity, x}], ε -> 0]/Pi

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$$\theta(x)=-\frac{i \lim _{\varepsilon \rightarrow 0} \int_{-\infty}^{\infty} \frac{e^{i t x}}{t-i \varepsilon} d t}{2 \pi}$$

1. HeavisideTheta[x] == (-I/2 Limit[Integrate[E^(I t x)/(t - I ε), {t, -Infinity, Infinity}], ε -> 0])/Pi

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$$\theta(x)=-\frac{i \lim _{\varepsilon \rightarrow 0} \int_{-\infty}^{\infty} \frac{e^{-i t x}}{t+i \varepsilon} d t}{2 \pi}$$

1. HeavisideTheta[x] == (-I/2 Limit[Integrate[E^(-I t x)/(t + I ε), {t, -Infinity, Infinity}], ε -> 0])/Pi

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$$\theta(x)=-\frac{i}{2 \pi} \int_{-i \infty+\gamma}^{i \infty+\gamma} \frac{(1+x)^{-s} \Gamma(-s)}{\Gamma(1-s)} d s \text { for }(\gamma<0 \text { and } x>-2)$$

1. HeavisideTheta[x] == (-I/2 Integrate[Gamma[-s]/((1 + x)^s Gamma[1 - s]), {s, -I Infinity + γ, I Infinity + γ}])/Pi /; γ < 0 && x > -2

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$$\theta(x)=-\frac{i}{2 \pi} \int_{-i \infty+\gamma}^{i \infty+\gamma} \frac{(1-x)^{-s} \Gamma(s)}{\Gamma(1+s)} d s \text { for }(0<\gamma \text { and } x<2)$$

1. HeavisideTheta[x] == (-I/2 Integrate[Gamma[s]/((1 - x)^s Gamma[1 + s]), {s, -I Infinity + γ, I Infinity + γ}])/Pi /; 0 < γ && x < 2

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