测试功能据/h1>
相关......据/h4>
- 结石据/S.pan>>据/S.pan>
这据S.T.R.ong>测试功能据/S.T.R.ong>(也称为据S.T.R.ong>欧拉的第一种据/S.T.R.ong>)在微积分和分析中是重要的,因为它紧密连接据一种href="//www.parkandroid.com/wiki/gamma-function/" class="wiki_link" title="伽玛功能" target="_blank">伽玛功能据/一种>,这本身就是阶乘函数的泛化。许多复杂积分可以减少到涉及β函数的表达式。据/p>
内容据/h4>
定义据/h2>
这据S.T.R.ong>测试功能据/S.T.R.ong>,表示据S.pan class="katex"> B.据/S.pan>(据/S.pan>X据/S.pan>那据/S.pan>y据/S.pan>)据/S.pan>,被定义为据/p>
B.据/S.pan>(据/S.pan>X据/S.pan>那据/S.pan>y据/S.pan>)据/S.pan>=据/S.pan>∫据/S.pan>0.据/S.pan>1据/S.pan>T.据/S.pan>X据/S.pan>-据/S.pan>1据/S.pan>(据/S.pan>1据/S.pan>-据/S.pan>T.据/S.pan>)据/S.pan>y据/S.pan>-据/S.pan>1据/S.pan>D.据/S.pan>T.据/S.pan>。据/S.pan>
这也是欧拉的第一类的整体。据/p>
对称据/h2>
Beta函数的对称性据/S.T.R.ong>
B.据/S.pan>(据/S.pan>X据/S.pan>那据/S.pan>y据/S.pan>)据/S.pan>=据/S.pan>B.据/S.pan>(据/S.pan>y据/S.pan>那据/S.pan>X据/S.pan>)据/S.pan>
由于确定积分的会聚属性据p>∫据/S.pan>0.据/S.pan>一种据/S.pan>F据/S.pan>(据/S.pan>T.据/S.pan>)据/S.pan>D.据/S.pan>T.据/S.pan>=据/S.pan>∫据/S.pan>0.据/S.pan>一种据/S.pan>F据/S.pan>(据/S.pan>一种据/S.pan>-据/S.pan>T.据/S.pan>)据/S.pan>D.据/S.pan>T.据/S.pan>那据/S.pan>
所以我们可以将上述整体重写为据/p>
B.据/S.pan>(据/S.pan>X据/S.pan>那据/S.pan>y据/S.pan>)据/S.pan>=据/S.pan>∫据/S.pan>0.据/S.pan>1据/S.pan>T.据/S.pan>y据/S.pan>-据/S.pan>1据/S.pan>(据/S.pan>1据/S.pan>-据/S.pan>T.据/S.pan>)据/S.pan>X据/S.pan>-据/S.pan>1据/S.pan>D.据/S.pan>T.据/S.pan>。据/S.pan>
因此,我们得到了测试函数是对称的,据S.pan class="katex"> B.据/S.pan>(据/S.pan>X据/S.pan>那据/S.pan>y据/S.pan>)据/S.pan>=据/S.pan>B.据/S.pan>(据/S.pan>y据/S.pan>那据/S.pan>X据/S.pan>)据/S.pan>。据/S.pan> □据/S.pan>
与伽玛功能相关据/h2>
我们有据/p>
B.据/S.pan>(据/S.pan>X据/S.pan>那据/S.pan>y据/S.pan>)据/S.pan>=据/S.pan>γ.据/S.pan>(据/S.pan>X据/S.pan>+据/S.pan>y据/S.pan>)据/S.pan>γ.据/S.pan>(据/S.pan>X据/S.pan>)据/S.pan>γ.据/S.pan>(据/S.pan>y据/S.pan>)据/S.pan>。据/S.pan>
对于正整数据S.pan class="katex"> X据/S.pan>和据S.pan class="katex"> y据/S.pan>,我们可以定义测试函数据/p>
B.据/S.pan>(据/S.pan>X据/S.pan>那据/S.pan>y据/S.pan>)据/S.pan>=据/S.pan>(据/S.pan>X据/S.pan>+据/S.pan>y据/S.pan>-据/S.pan>1据/S.pan>)据/S.pan>!!据/S.pan>(据/S.pan>X据/S.pan>-据/S.pan>1据/S.pan>)据/S.pan>!!据/S.pan>(据/S.pan>y据/S.pan>-据/S.pan>1据/S.pan>)据/S.pan>!!据/S.pan>。据/S.pan>
回想起伽玛函数的定义据/p>
γ.据/S.pan>(据/S.pan>S.据/S.pan>)据/S.pan>=据/S.pan>∫据/S.pan>0.据/S.pan>∞据/S.pan>X据/S.pan>S.据/S.pan>-据/S.pan>1据/S.pan>E.据/S.pan>-据/S.pan>X据/S.pan>D.据/S.pan>X据/S.pan>。据/S.pan>
现在人们可以写据/p>
γ.据/S.pan>(据/S.pan>m据/S.pan>)据/S.pan>γ.据/S.pan>(据/S.pan>N.据/S.pan>)据/S.pan>=据/S.pan>∫据/S.pan>0.据/S.pan>∞据/S.pan>X据/S.pan>m据/S.pan>-据/S.pan>1据/S.pan>E.据/S.pan>-据/S.pan>X据/S.pan>D.据/S.pan>X据/S.pan>∫据/S.pan>0.据/S.pan>∞据/S.pan>y据/S.pan>N.据/S.pan>-据/S.pan>1据/S.pan>E.据/S.pan>-据/S.pan>y据/S.pan>D.据/S.pan>y据/S.pan>。据/S.pan>
然后我们可以将其重写为双积分:据/p>
γ.据/S.pan>(据/S.pan>m据/S.pan>)据/S.pan>γ.据/S.pan>(据/S.pan>N.据/S.pan>)据/S.pan>=据/S.pan>∫据/S.pan>0.据/S.pan>∞据/S.pan>∫据/S.pan>0.据/S.pan>∞据/S.pan>X据/S.pan>m据/S.pan>-据/S.pan>1据/S.pan>y据/S.pan>N.据/S.pan>-据/S.pan>1据/S.pan>E.据/S.pan>-据/S.pan>(据/S.pan>X据/S.pan>+据/S.pan>y据/S.pan>)据/S.pan>D.据/S.pan>X据/S.pan>D.据/S.pan>y据/S.pan>。据/S.pan>
应用替代据S.pan class="katex"> X据/S.pan>=据/S.pan>V.据/S.pan>T.据/S.pan>和据S.pan class="katex"> y据/S.pan>=据/S.pan>V.据/S.pan>(据/S.pan>1据/S.pan>-据/S.pan>T.据/S.pan>)据/S.pan>那据/S.pan>我们有据/p>
γ.据/S.pan>(据/S.pan>m据/S.pan>)据/S.pan>γ.据/S.pan>(据/S.pan>N.据/S.pan>)据/S.pan>=据/S.pan>∫据/S.pan>0.据/S.pan>1据/S.pan>T.据/S.pan>m据/S.pan>-据/S.pan>1据/S.pan>(据/S.pan>1据/S.pan>-据/S.pan>T.据/S.pan>)据/S.pan>N.据/S.pan>-据/S.pan>1据/S.pan>D.据/S.pan>T.据/S.pan>∫据/S.pan>0.据/S.pan>∞据/S.pan>V.据/S.pan>m据/S.pan>+据/S.pan>N.据/S.pan>-据/S.pan>1据/S.pan>E.据/S.pan>-据/S.pan>V.据/S.pan>D.据/S.pan>V.据/S.pan>。据/S.pan>
我们使用了伽玛和β函数的定义据/p>
γ.据/S.pan>(据/S.pan>m据/S.pan>)据/S.pan>γ.据/S.pan>(据/S.pan>N.据/S.pan>)据/S.pan>=据/S.pan>B.据/S.pan>(据/S.pan>m据/S.pan>那据/S.pan>N.据/S.pan>)据/S.pan>γ.据/S.pan>(据/S.pan>m据/S.pan>+据/S.pan>N.据/S.pan>)据/S.pan>。据/S.pan>
因此证明。据S.pan class="katex"> □据/S.pan>
计算据S.pan class="katex"> B.据/S.pan>(据/S.pan>5.据/S.pan>那据/S.pan>7.据/S.pan>)据/S.pan>。据/S.pan>
如果我们通过对beta函数的定义进行计算来计算据S.pan class="katex"> B.据/S.pan>(据/S.pan>5.据/S.pan>那据/S.pan>7.据/S.pan>)据/S.pan>,我们将必须解决以下积分据/p>
B.据/S.pan>(据/S.pan>5.据/S.pan>那据/S.pan>7.据/S.pan>)据/S.pan>=据/S.pan>∫据/S.pan>0.据/S.pan>1据/S.pan>T.据/S.pan>4.据/S.pan>(据/S.pan>1据/S.pan>-据/S.pan>T.据/S.pan>)据/S.pan>6.据/S.pan>D.据/S.pan>T.据/S.pan>那据/S.pan>
这是一个非常繁琐的工作。以下是,当Beta函数与伽马功能的关系方面有方便:据/p>
B.据/S.pan>(据/S.pan>5.据/S.pan>那据/S.pan>7.据/S.pan>)据/S.pan>=据/S.pan>1据/S.pan>1据/S.pan>!!据/S.pan>4.据/S.pan>!!据/S.pan>6.据/S.pan>!!据/S.pan>=据/S.pan>1据/S.pan>1据/S.pan>×据/S.pan>1据/S.pan>0.据/S.pan>×据/S.pan>⋯据/S.pan>×据/S.pan>7.据/S.pan>4.据/S.pan>!!据/S.pan>=据/S.pan>2据/S.pan>3.据/S.pan>1据/S.pan>0.据/S.pan>1据/S.pan>。据/S.pan>□据/S.pan>
测试函数的三角形表示据/h2>
B.据/S.pan>(据/S.pan>X据/S.pan>那据/S.pan>y据/S.pan>)据/S.pan>=据/S.pan>∫据/S.pan>0.据/S.pan>π据/S.pan>/据/S.pan>2据/S.pan>2据/S.pan>罪据/S.pan>2据/S.pan>X据/S.pan>-据/S.pan>1据/S.pan>(据/S.pan>T.据/S.pan>)据/S.pan>COS.据/S.pan>2据/S.pan>y据/S.pan>-据/S.pan>1据/S.pan>(据/S.pan>T.据/S.pan>)据/S.pan>D.据/S.pan>T.据/S.pan>
考虑据/p>
B.据/S.pan>(据/S.pan>X据/S.pan>那据/S.pan>y据/S.pan>)据/S.pan>=据/S.pan>∫据/S.pan>0.据/S.pan>1据/S.pan>你据/S.pan>y据/S.pan>-据/S.pan>1据/S.pan>(据/S.pan>1据/S.pan>-据/S.pan>你据/S.pan>)据/S.pan>X据/S.pan>-据/S.pan>1据/S.pan>D.据/S.pan>你据/S.pan>。据/S.pan>
使用替代据S.pan class="katex"> 你据/S.pan>=据/S.pan>罪据/S.pan>2据/S.pan>(据/S.pan>T.据/S.pan>)据/S.pan>→据/S.pan>D.据/S.pan>你据/S.pan>=据/S.pan>2据/S.pan>罪据/S.pan>(据/S.pan>T.据/S.pan>)据/S.pan>COS.据/S.pan>(据/S.pan>T.据/S.pan>)据/S.pan>和据S.pan class="katex"> |据/S.pan>|据/S.pan>|据/S.pan>0.据/S.pan>1据/S.pan>→据/S.pan>|据/S.pan>|据/S.pan>|据/S.pan>0.据/S.pan>π据/S.pan>/据/S.pan>2据/S.pan>, 我们有据/p>
B.据/S.pan>(据/S.pan>X据/S.pan>那据/S.pan>y据/S.pan>)据/S.pan>=据/S.pan>∫据/S.pan>0.据/S.pan>π据/S.pan>/据/S.pan>2据/S.pan>2据/S.pan>罪据/S.pan>2据/S.pan>X据/S.pan>-据/S.pan>1据/S.pan>(据/S.pan>T.据/S.pan>)据/S.pan>COS.据/S.pan>2据/S.pan>y据/S.pan>-据/S.pan>1据/S.pan>(据/S.pan>T.据/S.pan>)据/S.pan>D.据/S.pan>T.据/S.pan>。据/S.pan>□据/S.pan>
找据/p>
∫据/S.pan>0.据/S.pan>π据/S.pan>/据/S.pan>2据/S.pan>罪据/S.pan>9.据/S.pan>(据/S.pan>X据/S.pan>)据/S.pan>COS.据/S.pan>(据/S.pan>X据/S.pan>)据/S.pan>D.据/S.pan>X据/S.pan>。据/S.pan>
给定的积分只是简单据/p>
2据/S.pan>1据/S.pan>B.据/S.pan>(据/S.pan>5.据/S.pan>那据/S.pan>1据/S.pan>)据/S.pan>=据/S.pan>2据/S.pan>1据/S.pan>⋅据/S.pan>γ.据/S.pan>(据/S.pan>6.据/S.pan>)据/S.pan>γ.据/S.pan>(据/S.pan>5.据/S.pan>)据/S.pan>γ.据/S.pan>(据/S.pan>1据/S.pan>)据/S.pan>=据/S.pan>2据/S.pan>1据/S.pan>⋅据/S.pan>5.据/S.pan>!!据/S.pan>4.据/S.pan>!!据/S.pan>=据/S.pan>1据/S.pan>0.据/S.pan>1据/S.pan>。据/S.pan>□据/S.pan>
评估据S.pan class="katex-display"> ∫据/S.pan>0.据/S.pan>2据/S.pan>π据/S.pan>(据/S.pan>COS.据/S.pan>X据/S.pan>+据/S.pan>罪据/S.pan>X据/S.pan>)据/S.pan>2据/S.pan>晒黑据/S.pan>X据/S.pan> D.据/S.pan>X据/S.pan>
使用替代据S.pan class="katex"> T.据/S.pan>=据/S.pan>晒黑据/S.pan>X据/S.pan>⟶据/S.pan>D.据/S.pan>X据/S.pan>=据/S.pan>COS.据/S.pan>2据/S.pan>(据/S.pan>X据/S.pan>)据/S.pan>D.据/S.pan>T.据/S.pan>所以,据/p>
∫据/S.pan>0.据/S.pan>2据/S.pan>π据/S.pan>(据/S.pan>COS.据/S.pan>X据/S.pan>+据/S.pan>罪据/S.pan>X据/S.pan>)据/S.pan>2据/S.pan>晒黑据/S.pan>X据/S.pan> D.据/S.pan>X据/S.pan>=据/S.pan>∫据/S.pan>0.据/S.pan>2据/S.pan>π据/S.pan>COS.据/S.pan>2据/S.pan>X据/S.pan>(据/S.pan>1据/S.pan>+据/S.pan>晒黑据/S.pan>X据/S.pan>)据/S.pan>2据/S.pan>晒黑据/S.pan>X据/S.pan> D.据/S.pan>X据/S.pan>=据/S.pan>∫据/S.pan>0.据/S.pan>∞据/S.pan>(据/S.pan>1据/S.pan>+据/S.pan>T.据/S.pan>)据/S.pan>2据/S.pan>T.据/S.pan>2据/S.pan>1据/S.pan>D.据/S.pan>T.据/S.pan>
并且通过使用Beta函数的备用定义,我们可以重写上述整数以适应定义据/p>
∫据/S.pan>0.据/S.pan>∞据/S.pan>(据/S.pan>1据/S.pan>+据/S.pan>T.据/S.pan>)据/S.pan>2据/S.pan>3.据/S.pan>+据/S.pan>T.据/S.pan>T.据/S.pan>2据/S.pan>3.据/S.pan>-据/S.pan>1据/S.pan>D.据/S.pan>T.据/S.pan>=据/S.pan>B.据/S.pan>(据/S.pan>2据/S.pan>3.据/S.pan>那据/S.pan>2据/S.pan>1据/S.pan>)据/S.pan>
然后通过切换到伽马函数据/p>
γ.据/S.pan>(据/S.pan>2据/S.pan>)据/S.pan>γ.据/S.pan>(据/S.pan>2据/S.pan>3.据/S.pan>)据/S.pan>γ.据/S.pan>(据/S.pan>2据/S.pan>1据/S.pan>)据/S.pan>=据/S.pan>1据/S.pan>!!据/S.pan>2据/S.pan>1据/S.pan>γ.据/S.pan>(据/S.pan>2据/S.pan>1据/S.pan>)据/S.pan>γ.据/S.pan>(据/S.pan>2据/S.pan>1据/S.pan>)据/S.pan>=据/S.pan>1据/S.pan>2据/S.pan>1据/S.pan>π据/S.pan> π据/S.pan> =据/S.pan>2据/S.pan>π据/S.pan>
复发关系据/h2>
测试功能的复发关系是给出的据/p>
B.据/S.pan>(据/S.pan>X据/S.pan>+据/S.pan>1据/S.pan>那据/S.pan>y据/S.pan>)据/S.pan>=据/S.pan>B.据/S.pan>(据/S.pan>X据/S.pan>那据/S.pan>y据/S.pan>)据/S.pan>X据/S.pan>+据/S.pan>y据/S.pan>X据/S.pan>。据/S.pan>
我们有据/p>
B.据/S.pan>(据/S.pan>X据/S.pan>+据/S.pan>1据/S.pan>那据/S.pan>y据/S.pan>)据/S.pan>=据/S.pan>(据/S.pan>X据/S.pan>+据/S.pan>y据/S.pan>)据/S.pan>!!据/S.pan>X据/S.pan>!!据/S.pan>(据/S.pan>y据/S.pan>-据/S.pan>1据/S.pan>)据/S.pan>!!据/S.pan>=据/S.pan>(据/S.pan>X据/S.pan>+据/S.pan>y据/S.pan>-据/S.pan>1据/S.pan>)据/S.pan>!!据/S.pan>(据/S.pan>X据/S.pan>-据/S.pan>1据/S.pan>)据/S.pan>!!据/S.pan>(据/S.pan>y据/S.pan>-据/S.pan>1据/S.pan>)据/S.pan>!!据/S.pan>×据/S.pan>X据/S.pan>+据/S.pan>y据/S.pan>X据/S.pan>=据/S.pan>B.据/S.pan>(据/S.pan>X据/S.pan>那据/S.pan>y据/S.pan>)据/S.pan>X据/S.pan>+据/S.pan>y据/S.pan>X据/S.pan>。据/S.pan>
从上面的关系和由于β函数的对称性,以下两种结果立即关注:据/p>
B.据/S.pan>(据/S.pan>X据/S.pan>那据/S.pan>y据/S.pan>+据/S.pan>1据/S.pan>)据/S.pan>B.据/S.pan>(据/S.pan>X据/S.pan>+据/S.pan>1据/S.pan>那据/S.pan>y据/S.pan>)据/S.pan>+据/S.pan>B.据/S.pan>(据/S.pan>X据/S.pan>那据/S.pan>y据/S.pan>+据/S.pan>1据/S.pan>)据/S.pan>=据/S.pan>B.据/S.pan>(据/S.pan>X据/S.pan>那据/S.pan>y据/S.pan>)据/S.pan>X据/S.pan>+据/S.pan>y据/S.pan>y据/S.pan>=据/S.pan>B.据/S.pan>(据/S.pan>X据/S.pan>那据/S.pan>y据/S.pan>)据/S.pan>。据/S.pan>□据/S.pan>
与中央二项系数的关系据/h2>
我们观察中央二项系数的倒数:据/p>
(据/S.pan>N.据/S.pan>2据/S.pan>N.据/S.pan>)据/S.pan>1据/S.pan>=据/S.pan>(据/S.pan>2据/S.pan>N.据/S.pan>)据/S.pan>!!据/S.pan>N.据/S.pan>!!据/S.pan>N.据/S.pan>!!据/S.pan>=据/S.pan>γ.据/S.pan>(据/S.pan>2据/S.pan>N.据/S.pan>+据/S.pan>1据/S.pan>)据/S.pan>γ.据/S.pan>(据/S.pan>N.据/S.pan>+据/S.pan>1据/S.pan>)据/S.pan>γ.据/S.pan>(据/S.pan>N.据/S.pan>+据/S.pan>1据/S.pan>)据/S.pan>=据/S.pan>γ.据/S.pan>(据/S.pan>2据/S.pan>N.据/S.pan>+据/S.pan>1据/S.pan>)据/S.pan>N.据/S.pan>γ.据/S.pan>(据/S.pan>N.据/S.pan>)据/S.pan>γ.据/S.pan>(据/S.pan>N.据/S.pan>+据/S.pan>1据/S.pan>)据/S.pan>=据/S.pan>N.据/S.pan>B.据/S.pan>(据/S.pan>N.据/S.pan>那据/S.pan>N.据/S.pan>+据/S.pan>1据/S.pan>)据/S.pan>。据/S.pan>
然后据/p>
B.据/S.pan>(据/S.pan>N.据/S.pan>那据/S.pan>N.据/S.pan>+据/S.pan>1据/S.pan>)据/S.pan>=据/S.pan>N.据/S.pan>(据/S.pan>N.据/S.pan>2据/S.pan>N.据/S.pan>)据/S.pan>1据/S.pan>。据/S.pan>
这是一个非常有用的关系,尤其是在解决求和时。据/p>
找据/p>
S.据/S.pan>=据/S.pan>N.据/S.pan>=据/S.pan>1据/S.pan>σ.据/S.pan>∞据/S.pan>N.据/S.pan>(据/S.pan>N.据/S.pan>2据/S.pan>N.据/S.pan>)据/S.pan>1据/S.pan>。据/S.pan>
使用与测试功能的关系,这是据/p>
S.据/S.pan>=据/S.pan>N.据/S.pan>=据/S.pan>1据/S.pan>σ.据/S.pan>∞据/S.pan>B.据/S.pan>(据/S.pan>N.据/S.pan>+据/S.pan>1据/S.pan>那据/S.pan>N.据/S.pan>)据/S.pan>=据/S.pan>N.据/S.pan>=据/S.pan>1据/S.pan>σ.据/S.pan>∞据/S.pan>∫据/S.pan>0.据/S.pan>1据/S.pan>X据/S.pan>N.据/S.pan>(据/S.pan>1据/S.pan>-据/S.pan>X据/S.pan>)据/S.pan>N.据/S.pan>-据/S.pan>1据/S.pan>D.据/S.pan>X据/S.pan>。据/S.pan>
我们可以由于积分的绝对融合而互换求和和积分迹象:据/p>
S.据/S.pan>=据/S.pan>∫据/S.pan>0.据/S.pan>1据/S.pan>N.据/S.pan>=据/S.pan>1据/S.pan>σ.据/S.pan>∞据/S.pan>X据/S.pan>N.据/S.pan>(据/S.pan>1据/S.pan>-据/S.pan>X据/S.pan>)据/S.pan>N.据/S.pan>-据/S.pan>1据/S.pan>D.据/S.pan>X据/S.pan>=据/S.pan>∫据/S.pan>0.据/S.pan>1据/S.pan>X据/S.pan>2据/S.pan>-据/S.pan>X据/S.pan>+据/S.pan>1据/S.pan>X据/S.pan>D.据/S.pan>X据/S.pan>。据/S.pan>
这据一种href="//www.parkandroid.com/wiki/geometric-progression-sum/" class="wiki_link" title="几何进展总和" target="_blank">几何进展总和据/一种>被使用了。现在我们可以轻松解决无限积分,然后放入极限。所以我们有据/p>
S.据/S.pan>=据/S.pan>9.据/S.pan>3.据/S.pan> π据/S.pan>。据/S.pan>□据/S.pan>
N.据/S.pan>=据/S.pan>1据/S.pan>σ.据/S.pan>∞据/S.pan>N.据/S.pan>2据/S.pan>(据/S.pan>2据/S.pan>N.据/S.pan>N.据/S.pan>)据/S.pan>1据/S.pan>=据/S.pan>B.据/S.pan>一种据/S.pan>ζ据/S.pan>(据/S.pan>C据/S.pan>)据/S.pan>
在上面的等式中,据S.pan class="katex">
一种据/S.pan>那据/S.pan>B.据/S.pan>那据/S.pan>C据/S.pan>是正整数,这样据S.pan class="katex">
一种据/S.pan>和据S.pan class="katex">
B.据/S.pan>是副仙。据/p>
找据S.pan class="katex">
一种据/S.pan>+据/S.pan>B.据/S.pan>+据/S.pan>C据/S.pan>。据/S.pan>
测试功能的衍生物据/h2>
测试功能的衍生物是解决一些积分的一种很好的方法:据/p>
∂据/S.pan>X据/S.pan>∂据/S.pan>B.据/S.pan>(据/S.pan>X据/S.pan>那据/S.pan>y据/S.pan>)据/S.pan>∂据/S.pan>X据/S.pan>∂据/S.pan>y据/S.pan>∂据/S.pan>2据/S.pan>B.据/S.pan>(据/S.pan>X据/S.pan>那据/S.pan>y据/S.pan>)据/S.pan>=据/S.pan>B.据/S.pan>(据/S.pan>X据/S.pan>那据/S.pan>y据/S.pan>)据/S.pan>(据/S.pan>ψ据/S.pan>(据/S.pan>X据/S.pan>)据/S.pan>-据/S.pan>ψ据/S.pan>(据/S.pan>X据/S.pan>+据/S.pan>y据/S.pan>)据/S.pan>)据/S.pan>=据/S.pan>B.据/S.pan>(据/S.pan>X据/S.pan>那据/S.pan>y据/S.pan>)据/S.pan>(据/S.pan>(据/S.pan>ψ据/S.pan>(据/S.pan>X据/S.pan>)据/S.pan>-据/S.pan>ψ据/S.pan>(据/S.pan>X据/S.pan>+据/S.pan>y据/S.pan>)据/S.pan>)据/S.pan>(据/S.pan>ψ据/S.pan>(据/S.pan>y据/S.pan>)据/S.pan>-据/S.pan>ψ据/S.pan>(据/S.pan>X据/S.pan>+据/S.pan>y据/S.pan>)据/S.pan>)据/S.pan>-据/S.pan>ψ据/S.pan>'据/S.pan>(据/S.pan>X据/S.pan>+据/S.pan>y据/S.pan>)据/S.pan>)据/S.pan>。据/S.pan>
让我们看看如何使用它,但首先阅读据一种href="//www.parkandroid.com/wiki/digamma-function/" class="wiki_link" title="Digamma功能" target="_blank">Digamma功能据/一种>维基。据/p>
找据/p>
∫据/S.pan>0.据/S.pan>1据/S.pan>LN.据/S.pan>(据/S.pan>T.据/S.pan>)据/S.pan>LN.据/S.pan>(据/S.pan>1据/S.pan>-据/S.pan>T.据/S.pan>)据/S.pan>D.据/S.pan>T.据/S.pan>。据/S.pan>
考虑据/p>
B.据/S.pan>(据/S.pan>X据/S.pan>那据/S.pan>y据/S.pan>)据/S.pan>=据/S.pan>∫据/S.pan>0.据/S.pan>1据/S.pan>T.据/S.pan>X据/S.pan>-据/S.pan>1据/S.pan>(据/S.pan>1据/S.pan>-据/S.pan>T.据/S.pan>)据/S.pan>y据/S.pan>-据/S.pan>1据/S.pan>D.据/S.pan>T.据/S.pan>。据/S.pan>
different据S.pan class="katex"> X据/S.pan>首先据S.pan class="katex"> y据/S.pan>要得到据/p>
∂据/S.pan>X据/S.pan>∂据/S.pan>y据/S.pan>∂据/S.pan>2据/S.pan>B.据/S.pan>(据/S.pan>X据/S.pan>那据/S.pan>y据/S.pan>)据/S.pan>=据/S.pan>∫据/S.pan>0.据/S.pan>1据/S.pan>LN.据/S.pan>(据/S.pan>T.据/S.pan>)据/S.pan>LN.据/S.pan>(据/S.pan>1据/S.pan>-据/S.pan>T.据/S.pan>)据/S.pan>T.据/S.pan>X据/S.pan>-据/S.pan>1据/S.pan>(据/S.pan>1据/S.pan>-据/S.pan>T.据/S.pan>)据/S.pan>y据/S.pan>-据/S.pan>1据/S.pan>D.据/S.pan>T.据/S.pan>。据/S.pan>
放据S.pan class="katex"> X据/S.pan>=据/S.pan>1据/S.pan>和据S.pan class="katex"> y据/S.pan>=据/S.pan>1据/S.pan>具有据/p>
B.据/S.pan>(据/S.pan>1据/S.pan>那据/S.pan>1据/S.pan>)据/S.pan>(据/S.pan>(据/S.pan>ψ据/S.pan>(据/S.pan>1据/S.pan>)据/S.pan>-据/S.pan>ψ据/S.pan>(据/S.pan>2据/S.pan>)据/S.pan>)据/S.pan>2据/S.pan>-据/S.pan>ψ据/S.pan>(据/S.pan>1据/S.pan>)据/S.pan>(据/S.pan>2据/S.pan>)据/S.pan>)据/S.pan>=据/S.pan>B.据/S.pan>(据/S.pan>1据/S.pan>那据/S.pan>1据/S.pan>)据/S.pan>(据/S.pan>(据/S.pan>-据/S.pan>1据/S.pan>)据/S.pan>2据/S.pan>-据/S.pan>ζ据/S.pan>(据/S.pan>2据/S.pan>)据/S.pan>+据/S.pan>1据/S.pan>)据/S.pan>=据/S.pan>2据/S.pan>-据/S.pan>6.据/S.pan>π据/S.pan>2据/S.pan>。据/S.pan>□据/S.pan>
近似据/h2>
使用据一种href="//www.parkandroid.com/wiki/stirlings-formula/" class="wiki_link" title="斯特林的惯例" target="_blank">斯特林的惯例据/一种>,我们可以轻松地定义测试函数的渐近近似值据/p>
B.据/S.pan>(据/S.pan>X据/S.pan>那据/S.pan>y据/S.pan>)据/S.pan>=据/S.pan>(据/S.pan>X据/S.pan>+据/S.pan>y据/S.pan>-据/S.pan>1据/S.pan>)据/S.pan>!!据/S.pan>(据/S.pan>X据/S.pan>-据/S.pan>1据/S.pan>)据/S.pan>!!据/S.pan>(据/S.pan>y据/S.pan>-据/S.pan>1据/S.pan>)据/S.pan>!!据/S.pan>〜据/S.pan>2据/S.pan>π据/S.pan> (据/S.pan>X据/S.pan>+据/S.pan>y据/S.pan>)据/S.pan>X据/S.pan>+据/S.pan>y据/S.pan>-据/S.pan>1据/S.pan>/据/S.pan>2据/S.pan>X据/S.pan>X据/S.pan>-据/S.pan>1据/S.pan>/据/S.pan>2据/S.pan>y据/S.pan>y据/S.pan>-据/S.pan>1据/S.pan>/据/S.pan>2据/S.pan>
为大据S.pan class="katex"> X据/S.pan>和大据S.pan class="katex"> y据/S.pan>。据/S.pan>
β发行据/h2>
在概率理论中,据S.T.R.ong>β发行据/S.T.R.ong>,写作据S.pan class="katex"> B.据/S.pan>E.据/S.pan>T.据/S.pan>一种据/S.pan>(据/S.pan>R.据/S.pan>那据/S.pan>S.据/S.pan>)据/S.pan>为了据S.pan class="katex"> R.据/S.pan>那据/S.pan>S.据/S.pan>>据/S.pan>0.据/S.pan>,由密度定义据/p>
B.据/S.pan>(据/S.pan>R.据/S.pan>那据/S.pan>S.据/S.pan>)据/S.pan>1据/S.pan>X据/S.pan>R.据/S.pan>-据/S.pan>1据/S.pan>(据/S.pan>1据/S.pan>-据/S.pan>X据/S.pan>)据/S.pan>S.据/S.pan>-据/S.pan>1据/S.pan>那据/S.pan>
在哪里据S.pan class="katex"> 0.据/S.pan>据据/S.pan>X据/S.pan>据据/S.pan>1据/S.pan>和据S.pan class="katex"> B.据/S.pan>(据/S.pan>R.据/S.pan>那据/S.pan>S.据/S.pan>)据/S.pan>=据S.pan class="katex"> ∫据/S.pan>0.据/S.pan>1据/S.pan>X据/S.pan>R.据/S.pan>-据/S.pan>1据/S.pan>(据/S.pan>1据/S.pan>-据/S.pan>X据/S.pan>)据/S.pan>S.据/S.pan>-据/S.pan>1据/S.pan>D.据/S.pan>X据/S.pan>由Beta函数定义。据/p>
B.据/S.pan>(据/S.pan>R.据/S.pan>那据/S.pan>S.据/S.pan>)据/S.pan>也被称为据S.T.R.ong>常规化常量据/S.T.R.ong>因为它使得整数等于1。据/p>
这据S.pan class="katex"> K.据/S.pan>TH.据/S.pan>-Order统计据S.pan class="katex"> N.据/S.pan>I.I.D.制服据S.pan class="katex"> (据/S.pan>0.据/S.pan>那据/S.pan>1据/S.pan>)据/S.pan>分布是模拟的据S.pan class="katex"> B.据/S.pan>E.据/S.pan>T.据/S.pan>一种据/S.pan>(据/S.pan>K.据/S.pan>那据/S.pan>N.据/S.pan>-据/S.pan>K.据/S.pan>+据/S.pan>1据/S.pan>)据/S.pan>。据/p>
在Mathematica的实施据/h2>
Beta函数可以在Mathematica中实现,如下:据/p>
1据/pre> |
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