函數內部也可以調用其他函數，如果一個函數自己調用了自己，就是遞歸函數。
我們以階乘為例1x2x3x4x······x(n-1)xn，用函數f(n)表示，有：f(n)=n!=1x2x3x4x······x(n-1)xn=(n-1)!xn=f(n-1)xn，有且只有n等於1時，屬於特殊情況，需要特殊處理：
<code>>>> def f(n):
\tif n==1:
\t\treturn 1
\treturn n*f(n-1)
>>> f(1)
1
>>> f(2)
2
>>> f(10)
3628800/<code>
對於f(5)，我們可以拆解程序的計算過程如下：
<code>判斷n!=0，
f(5)=5*f(4)
f(5)=5*(4*f(3))
f(5)=5*(4*(3*f(2)))
f(5)=5*(4*(3*(2*f(1))))
f(5)=5*(4*(3*(2*1)))
f(5)=5*(4*(3*2))
f(5)=5*(4*6)
f(5)=5*24
f(5)=120/<code>
基本所有的遞歸函數都可以寫成循環形式，但從邏輯上遞歸函數更加清晰、簡單。使用遞歸函數到防止棧溢出，函數調用是通過棧（stack）這種數據結構實現的，每當進入一個函數調用，棧就會加一層棧幀，每當函數返回，棧就會減一層棧幀。由於棧的大小不是無限的，所以，遞歸調用的次數過多，會導致棧溢出。我們試試f(100)、f(1000)和f（10000）
<code>>>> f(100)
93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000
>>> f(1000)
402387260077093773543702433923003985719374864210714632543799910429938512398629020592044208486969404800479988610197196058631666872994808558901323829669944590997424504087073759918823627727188732519779505950995276120874975462497043601418278094646496291056393887437886487337119181045825783647849977012476632889835955735432513185323958463075557409114262417474349347553428646576611667797396668820291207379143853719588249808126867838374559731746136085379534524221586593201928090878297308431392844403281231558611036976801357304216168747609675871348312025478589320767169132448426236131412508780208000261683151027341827977704784635868170164365024153691398281264810213092761244896359928705114964975419909342221566832572080821333186116811553615836546984046708975602900950537616475847728421889679646244945160765353408198901385442487984959953319101723355556602139450399736280750137837615307127761926849034352625200015888535147331611702103968175921510907788019393178114194545257223865541461062892187960223838971476088506276862967146674697562911234082439208160153780889893964518263243671616762179168909779911903754031274622289988005195444414282012187361745992642956581746628302955570299024324153181617210465832036786906117260158783520751516284225540265170483304226143974286933061690897968482590125458327168226458066526769958652682272807075781391858178889652208164348344825993266043367660176999612831860788386150279465955131156552036093988180612138558600301435694527224206344631797460594682573103790084024432438465657245014402821885252470935190620929023136493273497565513958720559654228749774011413346962715422845862377387538230483865688976461927383814900140767310446640259899490222221765904339901886018566526485061799702356193897017860040811889729918311021171229845901641921068884387121855646124960798722908519296819372388642614839657382291123125024186649353143970137428531926649875337218940694281434118520158014123344828015051399694290153483077644569099073152433278288269864602789864321139083506217095002597389863554277196742822248757586765752344220207573630569498825087968928162753848863396909959826280956121450994871701244516461260379029309120889086942028510640182154399457156805941872748998094254742173582401063677404595741785160829230135358081840096996372524230560855903700624271243416909004153690105933983835777939410970027753472000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
>>> f(10000)
Traceback (most recent call last):
File "<pyshell>", line 1, in <module>
f(10000)
File "<pyshell>", line 4, in f
return n*f(n-1)
File "<pyshell>", line 4, in f
return n*f(n-1)
File "<pyshell>", line 4, in f
return n*f(n-1)
[Previous line repeated 1021 more times]
File "<pyshell>", line 2, in f
if n==1:
RecursionError: maximum recursion depth exceeded in comparison/<pyshell>/<pyshell>/<pyshell>/<pyshell>/<module>/<pyshell>/<code>
如何解決溢出的問題呢？這就需要尾遞歸來優化。尾遞歸就是函數返回時調用自身，而且return不能使用表達式。對於編譯器或解釋器而言，每次遞歸實際上只使用了一個棧幀，就不會出現溢出了。我們來看下：
<code>>>> def f(n):
\treturn f2(n, 1)
>>> def f2(n2, m):
\tif n2 == 1:
\t\treturn m
\treturn f2(n2 - 1,n2 * m)
>>> f(5)
120/<code>
這裡相當於建立了一箇中間函數，每一次return f2(n2-1 , n2 * m)時，f2裡面的參數其實已經計算好了，一直使用的都是
<code>我們輸入f(5)
==>f2(5,1)
==>f2(4,5)
==>f2(3,20)
==>f2(2,60)
==>f2(1,120)
==>120/<code>
然而，實際上，我們一廂情願的希望這樣可以解決爆棧的問題，但是尾遞歸依然還是遞歸函數，不做優化一樣會爆棧，因為多數編程語言包括Python都沒有針對尾遞歸做優化，即使我們把函數改為尾遞歸，該爆棧一樣爆棧
<code>>>> f(10000)
Traceback (most recent call last):
File "<pyshell>", line 1, in <module>
f(10000)
File "<pyshell>", line 2, in f
return f2(n, 1)
File "<pyshell>", line 4, in f2
return f2(n2 - 1,n2 * m)
File "<pyshell>", line 4, in f2
return f2(n2 - 1,n2 * m)
File "<pyshell>", line 4, in f2
return f2(n2 - 1,n2 * m)
[Previous line repeated 1020 more times]
File "<pyshell>", line 2, in f2
if n2 == 1:
RecursionError: maximum recursion depth exceeded in comparison/<pyshell>/<pyshell>/<pyshell>/<pyshell>/<pyshell>/<module>/<pyshell>/<code>
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