1004 CHAPTER 28. HAUSDORFF MEASURE
28.4.1 A Formula For α (n)
What is α(n)? Recall the gamma function which makes sense for all p > 0.
Γ(p)≡∫
∞
0e−tt p−1dt.
Lemma 28.4.1 The following identities hold.
pΓ(p) = Γ(p+1),
Γ(p)Γ(q) =(∫ 1
0xp−1(1− x)q−1dx
)Γ(p+q),
Γ
(12
)=√
π
Proof: Using integration by parts,
Γ(p+1) =∫
∞
0e−tt pdt =−e−tt p|∞0 + p
∫∞
0e−tt p−1dt
= pΓ(p)
Next
Γ(p)Γ(q) =∫
∞
0e−tt p−1dt
∫∞
0e−ssq−1ds
=∫
∞
0
∫∞
0e−(t+s)t p−1sq−1dtds
=∫
∞
0
∫∞
se−u (u− s)p−1 sq−1duds
=∫
∞
0
∫ u
0e−u (u− s)p−1 sq−1dsdu
=∫
∞
0
∫ 1
0e−u (u−ux)p−1 (ux)q−1 udxdu
=∫
∞
0
∫ 1
0e−uup+q−1 (1− x)p−1 xq−1dxdu
= Γ(p+q)(∫ 1
0xp−1(1− x)q−1dx
).
It remains to find Γ( 1
2
).
Γ
(12
)=∫
∞
0e−tt−1/2dt =
∫∞
0e−u2 1
u2udu = 2
∫∞
0e−u2
du
Now (∫∞
0e−x2
dx)2
=∫
∞
0e−x2
dx∫
∞
0e−y2
dy =∫
∞
0
∫∞
0e−(x2+y2)dxdy
=∫
∞
0
∫π/2
0e−r2
rdθdr =14
π
100428.4.1 A Formula For a (7)CHAPTER 28. HAUSDORFF MEASUREWhat is (7)? Recall the gamma function which makes sense for all p > 0.r(p)= | ett? dt.0Lemma 28.4.1 The following identities hold.I(p)P(q)pl (p) =T (p+),(fax tax) P40)Proof: Using integration by parts,NextIt remains to find °(3)-r(3)-[ [ceva = | eS audu=2 | e"2 0 Uu 0NowT(p+1)(/2“dx:etl dt| e'tPdt =—e 't? | +p0 0pl (p)e"(u—s)?—| st! duds—s)P~! st ldsduux)?! (ux)?! udxdu“en typti! (1-xP! xl dxduT(p+q) (fa ~»)""as)[erafera-[ fece2 1 rd@dr = —er r 4— (Pty )dxdy