n=0.5的階乘
當n=0.5時,![n!=\frac{\sqrt{\pi}}{2}](/img/a/bf7/nBnauMWZ2czMygDM0UTO0MTM5MWYlNWOyEzNkhTY0MDM2MWNxMzYiVjM4kzLtVGdp9yYpB3LltWahJ2Lt92YuUHZpFmYuM3b09GawlGauI2LvoDc0RHa.jpg)
將n代入下面的公式。
![n!=\gamma\left(n+1\right)](/img/2/474/nBnauAzY5YmZ3YjM2Q2N4IGOzEmY1AjM1ImYkRDZihzM3UmMkVWY0QjYiR2LtVGdp9yYpB3LltWahJ2Lt92YuUHZpFmYuM3b09GawlGauE2LvoDc0RHa.jpg)
![\gamma\left(x\right)](/img/6/f84/nBnauETMycjZkVWN4EGNzAjNlVjY2ATMmVzM3UmZwUWNmRmM5ITYzYWMxY2LtVGdp9yYpB3LltWahJ2Lt92YuUHZpFmYuM3b09GawlGauM2LvoDc0RHa.jpg)
遞推公式:
![\gamma\left(s+1\right)=s\gamma\left(s\right)](/img/8/5c1/nBnauM2MkRGNhBjMhRjZ0IjMmZzY0MWO2YjNlJmZ1UDZ2Q2MlVWO1cTO3IzLtVGdp9yYpB3LltWahJ2Lt92YuUHZpFmYuM3b09GawlGauU2LvoDc0RHa.jpg)
余元公式:
![\gamma\left(s\right)\gamma\left(1-s\right)=\pi\div\sin\left(s\pi\right)](/img/2/ca0/nBnauMGO2MGZzATNhZGOwMWZyEWNkVGNlV2YiBDZyYzNyMjNzETY4MWZ4E2LtVGdp9yYpB3LltWahJ2Lt92YuUHZpFmYuM3b09GawlGauQ2LvoDc0RHa.jpg)
第一步:
![0.5!=\gamma\left(1.5\right)](/img/a/e1f/nBnaugTMlRjZiN2Y4UTMkJDOllDN4EWM4gDOygTNykTZ0YTN0UDZkNjM4AzLtVGdp9yYpB3LltWahJ2Lt92YuUHZpFmYuM3b09GawlGauI2LvoDc0RHa.jpg)
第二步:
![\gamma\left(1.5\right)=0.5\times\gamma\left(0.5\right)](/img/4/514/nBnauIzMhRTOwEDZ4UjM5UGN0UTZwQzN4I2YmRTN5QzMxI2YxIGZ4YGZmVzLtVGdp9yYpB3LltWahJ2Lt92YuUHZpFmYuM3b09GawlGauY2LvoDc0RHa.jpg)
第三步:
![\gamma\left(0.5\right)\gamma\left(0.5\right)=\pi\div\sin\left(0.5\times\pi\right)=\pi\div1=\pi](/img/2/87c/nBnauMzY4MDNjNGZlZDZhVDZ5ETZhRTY2IGNmVWMkdjMjRTY0kzN4Y2M1Q2LtVGdp9yYpB3LltWahJ2Lt92YuUHZpFmYuM3b09GawlGauc2LvoDc0RHa.jpg)
第四步:
![\sqrt{\pi}\approx1.77245](/img/3/a1d/nBnauEjZklDMxMGO5Q2Y3M2YxE2N4QDO1cTN2YzMiRTOwYzNmVmYjRWYwMzLtVGdp9yYpB3LltWahJ2Lt92YuUHZpFmYuM3b09GawlGauI2LvoDc0RHa.jpg)
![\gamma\left(0.5\right)\approx1.77245](/img/6/ae2/nBnaucDOiNDNmZGMiFjMkVWMmdDO5UGMkdDNhZ2M0QGO4YDZ1QjM5MDMkZ2LtVGdp9yYpB3LltWahJ2Lt92YuUHZpFmYuM3b09GawlGauQ2LvoDc0RHa.jpg)
第五步:
![\gamma\left(1.5\right)=0.5\times\gamma\left(0.5\right)\approx0.5\times1.77245=0.886225](/img/a/4c7/nBnauYjZhljNzEGN5AjN3YWZ5M2NmFWOzIjZjJTYwQ2NiJjYygTO1QzMjF2LtVGdp9yYpB3LltWahJ2Lt92YuUHZpFmYuM3b09GawlGauQ2LvoDc0RHa.jpg)
而
![0.886225\approx\frac{\sqrt{\pi}}{2}](/img/7/0fe/nBnauQmZ1ITZxM2NyMGNhRTO1gDN2UzY2UzY2UDNykzMwMmZkZ2MjV2YyM2LtVGdp9yYpB3LltWahJ2Lt92YuUHZpFmYuM3b09GawlGauM2LvoDc0RHa.jpg)
![\frac{\sqrt{\pi}}{2}](/img/6/2d7/nBnauUWZ3cDN1gDNzEjYjFjYmhDN4YGN3YzYiBjNjVTMzMmY1ITO5UmZxU2LtVGdp9yYpB3LltWahJ2Lt92YuUHZpFmYuM3b09GawlGauE2LvoDc0RHa.jpg)
n為正數的廣義階乘
n為正數時,上面講的計算廣義階乘的公式還正確嗎?那還是讓我們來算算吧。例如:n=1.4
第一步:
![1.4!=\gamma\left(2.4\right)](/img/8/547/nBnauYWO5QmYmJjYiZWN1QmNmNDNiRjZ3kDNhR2MxUTYmhDMjVGMhZGOwkzLtVGdp9yYpB3LltWahJ2Lt92YuUHZpFmYuM3b09GawlGauM2LvoDc0RHa.jpg)
![階乘(未知數為2.5!)](/img/a/b99/nBnauM3X3UTM3ETO5QjN3ATN5MTM2IDM1gjN5MTNwAzMxAzL0YzL0EzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
第二步:
![\gamma\left(2.4\right)=1.4\times\gamma\left(1.4\right)](/img/6/fab/nBnaucjZycjZkZWN4EGNzAjNlVjY2QGMkZ2Y3UmZwUWNmRmM5ITYzYWMxY2LtVGdp9yYpB3LltWahJ2Lt92YuUHZpFmYuM3b09GawlGauY2LvoDc0RHa.jpg)
第三步:
![\gamma\left(1.4\right)=0.4\times\gamma\left(0.4\right)](/img/c/cef/nBnauQjZ2MGZzATNhZGOwMWZyEWNkFzN2QGZiBDZyYzNyMjNzETY4MWZ4E2LtVGdp9yYpB3LltWahJ2Lt92YuUHZpFmYuM3b09GawlGauM2LvoDc0RHa.jpg)
第四步:查閱伽瑪函式表後,
![\gamma\left(0.4\right)\approx2.21816](/img/8/257/nBnauQDMiJmY5QzM0MDOyEmM2Q2N3QzYjNzYwEmM3QWOmlDZmBTM1MGNiFzLtVGdp9yYpB3LltWahJ2Lt92YuUHZpFmYuM3b09GawlGauU2LvoDc0RHa.jpg)
第五步:
![\gamma\left(1.4\right)\approx0.4\times2.21816=0.887264](/img/5/9f7/nBnauIzNygDNzQWY0QTMhRGZ1kTO3AzNjhTO4gzM1YDM4E2YldDMiNTMykzLtVGdp9yYpB3LltWahJ2Lt92YuUHZpFmYuM3b09GawlGauY2LvoDc0RHa.jpg)
第六步:
![\gamma\left(2.4\right)\approx1.4\times0.887264=1.2421696](/img/5/e6a/nBnauUzYxMDNygDM2EDMkZjNwUzNhdTNkdjZjJ2YxgTYkFTN0MGZkVmNjF2LtVGdp9yYpB3LltWahJ2Lt92YuUHZpFmYuM3b09GawlGauc2LvoDc0RHa.jpg)
所以說,1.2421696約等於1.4!的得數(用直等號就與1.4!的得數更接近了)。
上面的公式是完全正確的。
n為負數的廣義階乘
如果n為負數,又會怎么樣?首先,增加一個公式:
Legendre公式:
![\gamma\left(s\right)\gamma\left(s+\frac{1}{2}\right)=\left(\sqrt{\pi}\right)\gamma\left(2s\right)\div2^{\left(2s-1\right)}](/img/9/e1d/nBnauYDOlZTMxYTZkRWNlVzM3gTOzQzY1ITZzATNilDOzkjYlJjZyEDZ3MzLtVGdp9yYpB3LltWahJ2Lt92YuUHZpFmYuM3b09GawlGauI2LvoDc0RHa.jpg)
可以說,正數的所有方法都可以用進去,就是注意有些地方+變成-,-變成+。