0＜x,y≤π/2，z1=cosx/siny+icosy/sinx，｜z1｜=2，z2=√x+√y，求 ｜z1-z2｜ 最大值

wintex

請問複數
註 聽說z1模長是根號2，答案較好看

elim

这个题目的以下解相当繁. 抛砖引玉, 希望高人出手.

波斯猫猫

题：0＜x,y≤π/2，z1=cosx/siny+icosy/sinx，｜z1｜=2，z2=√x+i√y，求 ｜z1-z2｜ 最大值.
思路：显然， 由条件有（√x.sin2x+√y.sin2y）/(sinx.siny)≥0，当且仅当x和y取最大值x=y=π/2时等号成立.所以，｜z1-z2｜＾2=x+y-（√x.sin2x+√y.sin2y）/(sinx.siny)+4≤π+4，即max｜z1-z2｜=√(π+4)(当且仅当x=y=π/2时).

elim

\(x=y=\pi/2\) 时|z_1|=0.

elim

我已经发现了 \(\min|z_1-z_2|\). 继续征求简单解法.

elim

原来的解看错了题. 已经更正.

http://mathchina.elinkage.net/cg ... =1&topic=1109#2

elim

这种问题denglongshan, 天山草等的机器辅助应该有优势.

王守恩

1，N[Maximize[{Abs[(Cos[x]/Sin[y] +  I Cos[y]/Sin[x]) - (Sqrt[x] + I Sqrt[y])],
  Abs[Cos[x]/Sin[y]+ I Cos[y]/Sin[x]] ==2, \[Pi]/2 > x >0, \[Pi]/2 > y >0}, {x, y}]
  {0.89051389493118270862428102070460822406566351540014,
  {x ->0.61547970867038734106746452252006440723823411999680,
   y ->0.61547970867038734106746460757260654538429220086944}}

2，N[Maximize[{Abs[(Cos[x]/Sin[y] +  I Cos[y]/Sin[x]) - (Sqrt[x] + I Sqrt[y])],
  Abs[Cos[x]/Sin[y] + I Cos[y]/Sin[x]] == 2, x > 0, \[Pi]/2 > y}, {x, y}], 50]
  {3.1657412419308591448013321130590557223861335987054,
  {x ->2.1471516019201252103939424128623587360795578675609,
   y ->0.33598712459363576875492470952370340328717775757410}}

3，N[Minimize[{Abs[(Cos[x]/Sin[y] +  I Cos[y]/Sin[x]) - (Sqrt[x] +  I Sqrt[y])],
  Abs[Cos[x]/Sin[y] + I Cos[y]/Sin[x]] == 2, x > 0, \[Pi]/2 > y}, {x, y}], 50]
  {1.4871440336543749847296280041847106770528113156768,
  {x ->0.91407849016242912996769283800428145538187035800063,
   y ->-2.7557391074202087488525728591994351761923765313840}}

4，N[Minimize[{Abs[(Cos[x]/Sin[y] +  I Cos[y]/Sin[x]) - (Sqrt[x] +  I Sqrt[y])],
   Abs[Cos[x]/Sin[y]+ I Cos[y]/Sin[x]] ==2, \[Pi]/2 > x >0, \[Pi]/2 > y >0}, {x, y}]
   {0.84311482008389357997112200870383357934737570176338,
   {x ->0.23602589794522812093816639091496331094549836058626,
    y ->1.1628651951026096057944891892867700496595713729350}}

elim

@王守恩，画出来看看？

王守恩

王守恩 发表于 2020-12-25 10:43
1，N[Maximize[{Abs[(Cos[x]/Sin[y] +  I Cos[y]/Sin[x]) - (Sqrt[x] + I Sqrt[y])],
  Abs[Cos[x]/Sin[y ...

1，N[Maximize[{Abs[(Cos[x]/Sin[y] + ( I Cos[y])/Sin[x]) - (Sqrt[x] +  I Sqrt[y])],
    Abs[Cos[x]/Sin[y] + ( I Cos[y])/Sin[x]] == 2, \[Pi]/2>x>0,\[Pi]/2>y>0},{x, y}]
  {0.89051389493118270862428102070460822406566351540014,
  {x ->0.61547970867038734106746452252006440723823411999680,
   y ->0.61547970867038734106746460757260654538429220086944}}

2，N[Maximize[{Sqrt[(Cos[x]/Sin[y])^2 + (Cos[y]/Sin[x])^2] - Sqrt[x + y],
   Sqrt[(Cos[x]/Sin[y])^2+(Cos[y]/Sin[x])^2]==2,\[Pi]/2>x>0,\[Pi]/2>y>0},{x, y}]
  {0.89051389493118270862428102070460822406566351540015,
  {x ->0.61547970867038734106746447807522429234534151565555,
   y ->0.61547970867038734106746465201744666027718480521069}}

3，N[Maximize[{Sqrt[(Cos[x]/Sin[x])^2 + (Cos[x]/Sin[x])^2] - Sqrt[x + x],
   Sqrt[(Cos[x]/Sin[x])^2 + (Cos[x]/Sin[x])^2] == 2, \[Pi]/2 > x > 0}, {x}], 50]
  {0.89051389493118270862428089685012470126530913391210,
  {x ->0.61547970867038734106746458912399368785517000467755}}

4，Maximize[{Sqrt[(Cos[x]/Sin[x])^2 + (Cos[x]/Sin[x])^2] - Sqrt[x + x],
    Sqrt[(Cos[x]/Sin[x])^2 + (Cos[x]/Sin[x])^2] == 2, \[Pi]/2 > x > 0}, {x}]
   {2 - Sqrt[2 ArcCot[Sqrt[2]]], {x -> ArcCot[Sqrt[2]]}}