和三角式有关的取值范围问题

manthanein

求下面这个式子的取值范围，n是正整数
\(\displaystyle \prod_{k=1}^{n} \sin{k\theta}\)

manthanein

我推测这个式子有最大值和最小值，且两者互为相反数，下面只考虑最大值
当k=1时，max=1
其余的通过作图：
当k=2时，max=0.76980
当k=3时，max=0.54874
当k逐渐增大时，原式的图像似乎越来越接近y=0

zeroieme

根据欧拉公式$ e^{ix} = \cos x + i\sin x $可转化为讨论等比数列和的虚部

manthanein

zeroieme 发表于 2016-5-3 23:56
根据欧拉公式$ e^{ix} = \cos x + i\sin x $可转化为讨论等比数列和的虚部

试过了，好像不行，没法化简

mathe

这么多项乘积趋向0是显然的

zeroieme

manthanein 发表于 2016-5-4 19:38
试过了，好像不行，没法化简

对不起，连乘看成连加了

wayne

http://mathworld.wolfram.com/Multiple-AngleFormulas.html
本质上是多项式的求最值问题：
Column[Table[{n,N[Maximize[{(1-x^2)^(Floor[n/2]/2) Product[ChebyshevT[k,x],{k,1,n,2}]
Product[ChebyshevU[k-1,x],{k,2,n,2}],-1<=x<=1},x],50]},{n,20}]]
答案：

{1,{1.0000000000000000000000000000000000000000000000000,{x->1.0000000000000000000000000000000000000000000000000}}}
{2,{0.76980035891950101934553170733594327419680233502684,{x->-0.81649658092772603273242802490196379732198249355222}}}
{3,{0.54873704637207574204910449043320477637566607000809,{x->-0.63493584582956231321705573769733195132075422795091}}}
{4,{0.37656404906080095144029710431717793916809229470337,{x->-0.51237169622875526922387851183238377168494304463874}}}
{5,{0.25256020377881647210138405322331682220257518019029,{x->0.42759305633358182517074478388444964277496112137109}}}
{6,{0.16680640609126432620315825387423223494144100242735,{x->-0.36620688635158696230589578294745354480618523415261}}}
{7,{0.10895757349003181879282057798983424820597084204858,{x->-0.31993731612726228869172856945724773450561103002848}}}
{8,{0.070579300691593439999172398828708314764030093697532,{x->-0.28390218008474968314302927747125526124545189702925}}}
{9,{0.045421620026365375837458920819309411401100748976263,{x->0.25508373905282455790392538351500009478085317824243}}}
{10,{0.029078324892112134166583815580149661488660716039134,{x->-0.23153099754906201096687878016875023371003885004198}}}
{11,{0.018535495560027094263799982710870816709810919974886,{x->-0.21193199374982475336498785799053614244618106859734}}}
{12,{0.011772627815547662932801873746836038780197519802053,{x->-0.19537418776725465598172768069840954302282425478019}}}
{13,{0.0074544167076104689401155459836510884445497798812334,{x->0.18120422283990602209753529786839485189996622186964}}}
{14,{0.0047077305932337080631678489838644671686342744921594,{x->-0.16894250650959042863608684290061614826862658247133}}}
{15,{0.0029663135030568058630944885847427153807417843336325,{x->-0.15822930553034076814107592743289423326335019484971}}}
{16,{0.0018653147416096561946092444023366816959376617472606,{x->-0.14878964739912023023905913861735503559537937445761}}}
{17,{0.0011708942481310910598168863793951381473362073618962,{x->0.14040982730266064723171797121140341493179651392464}}}
{18,{0.00073383421469409406206408925277927123608373573327680,{x->-0.13292129361862446784474089853013567369833532993757}}}
{19,{0.00045926578848066167834510765541399113019723682199512,{x->-0.12618935120140631468522565634643654299525307724172}}}
{20,{0.00028706263757777999096568476999417626994718747410771,{x->-0.12010508629681618708420843986092721344350582870503}}}

lsr314

令$f(x)=sinx*sin(2x)*……*sin(nx)$
$f'(x)=cosx*f(x)/sin(x)+……+ncos(nx)*f(x)/sin(nx)$
由于f(x)/sin(kx)总是有意义的，所以f(x)在f'(x)=0时取极值，排除f(x)=0，得到极值条件：
$cot(x)+2cot(2x)+……+ncot(nx)=0$
这个方程的最小正数解对应的是f(x)的最大值m.
有趣的是，当n是4的倍数时，f(x)的最大值和最小值并不是互为相反数，比如n=4时，最大值是0.376564，最小值是-0.146633,函数在$(0,2\pi)$内的图像如下：

n=19时，最大值与最小值也不是互为相反数，那么，哪些n对应的最大值和最小值绝对值不相等呢？

补充内容 (2016-5-18 09:33):
n是奇数时最大值和最小值是互为相反数的

wayne

lsr314 发表于 2016-5-9 12:51
令$f(x)=sinx*sin(2x)*……*sin(nx)$
$f'(x)=cosx*f(x)/sin(x)+……+ncos(nx)*f(x)/sin(nx)$
由于f(x)/si ...

n为奇数时，最大值和最小值是互为相反数的。
n为偶数时，不是互为相反数。

1. {1,1.0000000000000000000000000000000000000000000000000,-1.0000000000000000000000000000000000000000000000000}
   
2. {2,0.76980035891950101934553170733594327419680233502684,0}
   
3. {3,0.54873704637207574204910449043320477637566607000809,-0.54873704637207574204910449043320477637566607000809}
   
4. {4,0.37656404906080095144029710431717793916809229470337,-0.14663292250781130166930215917513726066472593245847}
   
5. {5,0.25256020377881647210138405322331682220257518019029,-0.25256020377881647210138405322331682220257518019029}
   
6. {6,0.16680640609126432620315825387423223494144100242735,-0.10974171829225730197227808351388375143436894200548}
   
7. {7,0.10895757349003181879282057798983424820597084204858,-0.10895757349003181879282057798983424820597084204858}
   
8. {8,0.070579300691593439999172398828708314764030093697532,-0.026562150510300222231098930414905714195694925703362}
   
9. {9,0.045421620026365375837458920819309411401100748976263,-0.045421620026365375837458920819309411401100748976263}
   
10. {10,0.029078324892112134166583815580149661488660716039134,-0.011540034509884791367786564114342086837218004691902}
    
11. {11,0.018535495560027094263799982710870816709810919974886,-0.018535495560027094263799982710870816709810919974886}
    
12. {12,0.011772627815547662932801873746836038780197519802053,-0.0031118481883156193840421931031327691020618968633329}
    
13. {13,0.0074544167076104689401155459836510884445497798812334,-0.0074544167076104689401155459836510884445497798812334}
    
14. {14,0.0047077305932337080631678489838644671686342744921594,-0.0010130586602265054833385086353763837977970796530575}
    
15. {15,0.0029663135030568058630944885847427153807417843336325,-0.0029663135030568058630944885847427153807417843336325}
    
16. {16,0.0018653147416096561946092444023366816959376617472606,-0.00036752388565867416666713749774603087138541791830274}
    
17. {17,0.0011708942481310910598168863793951381473362073618962,-0.0011708942481310910598168863793951381473362073618962}
    
18. {18,0.00073383421469409406206408925277927123608373573327680,-0.000096342002751112705488520424670093930928626289819656}
    
19. {19,0.00045926578848066167834510765541399113019723682199512,-0.00045926578848066167834510765541399113019723682199512}
    
20. {20,0.00028706263757777999096568476999417626994718747410771,-0.000039213505579498052822098633217408338820004414255732}
    

复制代码

lsr314

wayne 发表于 2016-5-9 17:51
n为奇数时，最大值和最小值是互为相反数的。
n为偶数时，不是互为相反数。

n=2的时候最小值怎么会是0呢？n=6的时候最小值是可以取到-0.1668的，其他的几个也是