34 lines
1011 B
Markdown
34 lines
1011 B
Markdown
#Math #Calculus
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# Extending the Factorial Function
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We know $n!$ has a restricted domain of $n \in \mathbb{N}$, but we want to extend this function to $n \in \mathbb{R}$. To do this, we define two basic properties for the gamma function:
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$$
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n\Gamma(n) = \Gamma(n + 1) \\
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\Gamma(n + 1) = n!, \space n\in \mathbb{N}
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$$
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# Derivation
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We know repeated differentiation can generate a factorial function, so we start by differentiating:
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$$
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\int _{0}^{\infty} e^{-ax} dx = \frac 1 a
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$$
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**Lebeniz Integral Rule** allows us to differentiate inside the integral, so by repeated differentiation with respect to $a$ and cancelling out the negative sign we get:
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$$
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\int _{0}^{\infty} xe^{-ax} dx = \frac 1 {a^2} \\
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\int _{0}^{\infty} x^2e^{-ax} dx = \frac 2 {a^3} \\
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\int _{0}^{\infty} x^ne^{-ax} dx = \frac {n!} {a^{n + 1}} \\
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$$
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Plugging $a = 1$ we get:
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$$
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\Gamma(n) = \int _{0}^{\infty} x^{n - 1} e^{-x} dx
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$$
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Plugging the definition into the above properties should affirm that this defines the gamma function. |