{
  "index": "1949-A-6",
  "type": "ANA",
  "tag": [
    "ANA",
    "ALG"
  ],
  "difficulty": "",
  "question": "\\begin{array}{l}\n\\text { 6. Prove that for every real or complex } \\boldsymbol{x}\\\\\n\\prod_{k=1}^{\\infty} \\frac{1+2 \\cos \\frac{2 x}{3^{k}}}{3}=\\frac{\\sin x}{x}\n\\end{array}",
  "solution": "Solution. The identity\n\\[\n\\sin 3 \\theta-\\sin \\theta=2 \\sin \\frac{1}{2}(3 \\theta-\\theta) \\cos \\frac{1}{2}(3 \\theta+\\theta)\n\\]\nleads to the identity\n\\[\n\\sin 3 \\theta=\\sin \\theta(1+2 \\cos 2 \\theta) .\n\\]\n\nHence for any \\( \\boldsymbol{x} \\),\n\\[\n\\begin{aligned}\n\\sin x & =\\sin (x / 3)(1+2 \\cos (2 x / 3)) \\\\\n& =\\sin (x / 9)(1+2 \\cos (2 x / 9))(1+2 \\cos (2 x / 3))\n\\end{aligned}\n\\]\nand after \\( \\boldsymbol{n} \\) iterations we have\n\\[\n\\sin x=\\sin \\left(x / 3^{n}\\right) \\prod_{k=1}^{n}\\left(1+2 \\cos \\left(2 x / 3^{k}\\right)\\right)\n\\]\n\nFor \\( x=0 \\), the right member of the required relation is not defined. However, if we interpret \\( (\\sin 0) / 0 \\) as \\( 1\\left(=\\lim _{x \\rightarrow 0} \\sin x / x\\right) \\), then the required equation is correct since every factor in the infinite product is 1.\n\nIf \\( x \\neq 0 \\), we divide (1) by \\( x \\) and rearrange to get\n\\[\n\\frac{\\sin x}{x}=\\frac{\\sin \\left(x / 3^{n}\\right)}{x / 3^{n}} \\prod_{k=1}^{n}\\left(\\frac{1+2 \\cos \\left(2 x / 3^{k}\\right)}{3}\\right) .\n\\]\n\nNow\n\\[\n\\lim _{n \\rightarrow \\infty} \\frac{\\sin \\left(x / 3^{n}\\right)}{x / 3^{n}}=1,\n\\]\nand it follows that\n\\[\n\\lim _{n \\rightarrow \\infty} \\prod_{k=1}^{n} \\frac{1+2 \\cos \\left(2 x / 3^{k}\\right)}{3}=\\frac{\\sin x}{x} .\n\\]\n\nRemark. Because of the special behavior of 0 in multiplication, the convergence of infinite products is not usually decided simply by the convergence of the sequence of partial products. Rather, it is required that only finitely many factors be zero and that the partial products of the others have a non-zero limit. (See, for example, Ahlfors, Complex Analysis, 2nd ed., McGraw-Hill, New York, 1966, p. 189.) The infinite product above converges in this restricted sense because for any fixed \\( \\boldsymbol{x} \\),\n\\[\n1+2 \\cos \\frac{2 x}{3^{k}}\n\\]\ndoes not vanish for large \\( k \\) and the above proof shows that\n\\[\n\\lim _{n \\rightarrow \\infty} \\prod_{k=t+1}^{n}\\left(1+2 \\cos \\frac{2 x}{3^{k}}\\right)=\\frac{\\sin \\left(x / 3^{t}\\right)}{x / 3^{t}} \\neq 0\n\\]\nfor large \\( t \\).",
  "vars": [
    "x",
    "k",
    "n",
    "t",
    "\\\\theta"
  ],
  "params": [],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "inputvar",
        "k": "termindex",
        "n": "levelnum",
        "t": "shiftidx",
        "\\theta": "anglevar"
      },
      "question": "\\begin{array}{l}\n\\text { 6. Prove that for every real or complex } \\boldsymbol{inputvar}\\\\\n\\prod_{termindex=1}^{\\infty} \\frac{1+2 \\cos \\frac{2 inputvar}{3^{termindex}}}{3}=\\frac{\\sin inputvar}{inputvar}\n\\end{array}",
      "solution": "Solution. The identity\n\\[\n\\sin 3 anglevar-\\sin anglevar=2 \\sin \\frac{1}{2}(3 anglevar-anglevar) \\cos \\frac{1}{2}(3 anglevar+anglevar)\n\\]\nleads to the identity\n\\[\n\\sin 3 anglevar=\\sin anglevar(1+2 \\cos 2 anglevar) .\n\\]\n\nHence for any \\( \\boldsymbol{inputvar} \\),\n\\[\n\\begin{aligned}\n\\sin inputvar & =\\sin (inputvar / 3)(1+2 \\cos (2 inputvar / 3)) \\\\\n& =\\sin (inputvar / 9)(1+2 \\cos (2 inputvar / 9))(1+2 \\cos (2 inputvar / 3))\n\\end{aligned}\n\\]\nand after \\( \\boldsymbol{levelnum} \\) iterations we have\n\\[\n\\sin inputvar=\\sin \\left(inputvar / 3^{levelnum}\\right) \\prod_{termindex=1}^{levelnum}\\left(1+2 \\cos \\left(2 inputvar / 3^{termindex}\\right)\\right)\n\\]\n\nFor \\( inputvar=0 \\), the right member of the required relation is not defined. However, if we interpret \\( (\\sin 0) / 0 \\) as \\( 1\\left(=\\lim _{inputvar \\rightarrow 0} \\sin inputvar / inputvar\\right) \\), then the required equation is correct since every factor in the infinite product is 1.\n\nIf \\( inputvar \\neq 0 \\), we divide (1) by \\( inputvar \\) and rearrange to get\n\\[\n\\frac{\\sin inputvar}{inputvar}=\\frac{\\sin \\left(inputvar / 3^{levelnum}\\right)}{inputvar / 3^{levelnum}} \\prod_{termindex=1}^{levelnum}\\left(\\frac{1+2 \\cos \\left(2 inputvar / 3^{termindex}\\right)}{3}\\right) .\n\\]\n\nNow\n\\[\n\\lim _{levelnum \\rightarrow \\infty} \\frac{\\sin \\left(inputvar / 3^{levelnum}\\right)}{inputvar / 3^{levelnum}}=1,\n\\]\nand it follows that\n\\[\n\\lim _{levelnum \\rightarrow \\infty} \\prod_{termindex=1}^{levelnum} \\frac{1+2 \\cos \\left(2 inputvar / 3^{termindex}\\right)}{3}=\\frac{\\sin inputvar}{inputvar} .\n\\]\n\nRemark. Because of the special behavior of 0 in multiplication, the convergence of infinite products is not usually decided simply by the convergence of the sequence of partial products. Rather, it is required that only finitely many factors be zero and that the partial products of the others have a non-zero limit. (See, for example, Ahlfors, Complex Analysis, 2nd ed., McGraw-Hill, New York, 1966, p. 189.) The infinite product above converges in this restricted sense because for any fixed \\( \\boldsymbol{inputvar} \\),\n\\[\n1+2 \\cos \\frac{2 inputvar}{3^{termindex}}\n\\]\ndoes not vanish for large \\( termindex \\) and the above proof shows that\n\\[\n\\lim _{levelnum \\rightarrow \\infty} \\prod_{termindex=shiftidx+1}^{levelnum}\\left(1+2 \\cos \\frac{2 inputvar}{3^{termindex}}\\right)=\\frac{\\sin \\left(inputvar / 3^{shiftidx}\\right)}{inputvar / 3^{shiftidx}} \\neq 0\n\\]\nfor large \\( shiftidx \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "mapleleaf",
        "k": "snowflake",
        "n": "sandcastle",
        "t": "moonlight",
        "\\theta": "butterfly"
      },
      "question": "\\begin{array}{l}\n\\text { 6. Prove that for every real or complex } \\boldsymbol{mapleleaf}\\\\\n\\prod_{snowflake=1}^{\\infty} \\frac{1+2 \\cos \\frac{2 mapleleaf}{3^{snowflake}}}{3}=\\frac{\\sin mapleleaf}{mapleleaf}\n\\end{array}",
      "solution": "Solution. The identity\n\\[\n\\sin 3 butterfly-\\sin butterfly=2 \\sin \\frac{1}{2}(3 butterfly-butterfly) \\cos \\frac{1}{2}(3 butterfly+butterfly)\n\\]\nleads to the identity\n\\[\n\\sin 3 butterfly=\\sin butterfly(1+2 \\cos 2 butterfly) .\n\\]\n\nHence for any \\( \\boldsymbol{mapleleaf} \\),\n\\[\n\\begin{aligned}\n\\sin mapleleaf & =\\sin (mapleleaf / 3)(1+2 \\cos (2 mapleleaf / 3)) \\\\\n& =\\sin (mapleleaf / 9)(1+2 \\cos (2 mapleleaf / 9))(1+2 \\cos (2 mapleleaf / 3))\n\\end{aligned}\n\\]\nand after \\( \\boldsymbol{sandcastle} \\) iterations we have\n\\[\n\\sin mapleleaf=\\sin \\left(mapleleaf / 3^{sandcastle}\\right) \\prod_{snowflake=1}^{sandcastle}\\left(1+2 \\cos \\left(2 mapleleaf / 3^{snowflake}\\right)\\right)\n\\]\n\nFor \\( mapleleaf=0 \\), the right member of the required relation is not defined. However, if we interpret \\( (\\sin 0) / 0 \\) as \\( 1\\left(=\\lim _{mapleleaf \\rightarrow 0} \\sin mapleleaf / mapleleaf\\right) \\), then the required equation is correct since every factor in the infinite product is 1.\n\nIf \\( mapleleaf \\neq 0 \\), we divide (1) by \\( mapleleaf \\) and rearrange to get\n\\[\n\\frac{\\sin mapleleaf}{mapleleaf}=\\frac{\\sin \\left(mapleleaf / 3^{sandcastle}\\right)}{mapleleaf / 3^{sandcastle}} \\prod_{snowflake=1}^{sandcastle}\\left(\\frac{1+2 \\cos \\left(2 mapleleaf / 3^{snowflake}\\right)}{3}\\right) .\n\\]\n\nNow\n\\[\n\\lim _{sandcastle \\rightarrow \\infty} \\frac{\\sin \\left(mapleleaf / 3^{sandcastle}\\right)}{mapleleaf / 3^{sandcastle}}=1,\n\\]\nand it follows that\n\\[\n\\lim _{sandcastle \\rightarrow \\infty} \\prod_{snowflake=1}^{sandcastle} \\frac{1+2 \\cos \\left(2 mapleleaf / 3^{snowflake}\\right)}{3}=\\frac{\\sin mapleleaf}{mapleleaf} .\n\\]\n\nRemark. Because of the special behavior of 0 in multiplication, the convergence of infinite products is not usually decided simply by the convergence of the sequence of partial products. Rather, it is required that only finitely many factors be zero and that the partial products of the others have a non-zero limit. (See, for example, Ahlfors, Complex Analysis, 2nd ed., McGraw-Hill, New York, 1966, p. 189.) The infinite product above converges in this restricted sense because for any fixed \\( \\boldsymbol{mapleleaf} \\),\n\\[\n1+2 \\cos \\frac{2 mapleleaf}{3^{snowflake}}\n\\]\ndoes not vanish for large \\( snowflake \\) and the above proof shows that\n\\[\n\\lim _{sandcastle \\rightarrow \\infty} \\prod_{snowflake=moonlight+1}^{sandcastle}\\left(1+2 \\cos \\frac{2 mapleleaf}{3^{snowflake}}\\right)=\\frac{\\sin \\left(mapleleaf / 3^{moonlight}\\right)}{mapleleaf / 3^{moonlight}} \\neq 0\n\\]\nfor large \\( moonlight \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "constantval",
        "k": "unindexed",
        "n": "continuous",
        "t": "finality",
        "\\theta": "straightline"
      },
      "question": "\\begin{array}{l}\n\\text { 6. Prove that for every real or complex } \\boldsymbol{constantval}\\\\\n\\prod_{unindexed=1}^{\\infty} \\frac{1+2 \\cos \\frac{2 constantval}{3^{unindexed}}}{3}=\\frac{\\sin constantval}{constantval}\n\\end{array}",
      "solution": "Solution. The identity\n\\[\n\\sin 3 straightline-\\sin straightline=2 \\sin \\frac{1}{2}(3 straightline-straightline) \\cos \\frac{1}{2}(3 straightline+straightline)\n\\]\nleads to the identity\n\\[\n\\sin 3 straightline=\\sin straightline(1+2 \\cos 2 straightline) .\n\\]\n\nHence for any \\( \\boldsymbol{constantval} \\),\n\\[\n\\begin{aligned}\n\\sin constantval & =\\sin (constantval / 3)(1+2 \\cos (2 constantval / 3)) \\\\\n& =\\sin (constantval / 9)(1+2 \\cos (2 constantval / 9))(1+2 \\cos (2 constantval / 3))\n\\end{aligned}\n\\]\nand after \\( \\boldsymbol{continuous} \\) iterations we have\n\\[\n\\sin constantval=\\sin \\left(constantval / 3^{continuous}\\right) \\prod_{unindexed=1}^{continuous}\\left(1+2 \\cos \\left(2 constantval / 3^{unindexed}\\right)\\right)\n\\]\n\nFor \\( constantval=0 \\), the right member of the required relation is not defined. However, if we interpret \\( (\\sin 0) / 0 \\) as \\( 1\\left(=\\lim _{constantval \\rightarrow 0} \\sin constantval / constantval\\right) \\), then the required equation is correct since every factor in the infinite product is 1.\n\nIf \\( constantval \\neq 0 \\), we divide (1) by \\( constantval \\) and rearrange to get\n\\[\n\\frac{\\sin constantval}{constantval}=\\frac{\\sin \\left(constantval / 3^{continuous}\\right)}{constantval / 3^{continuous}} \\prod_{unindexed=1}^{continuous}\\left(\\frac{1+2 \\cos \\left(2 constantval / 3^{unindexed}\\right)}{3}\\right) .\n\\]\n\nNow\n\\[\n\\lim _{continuous \\rightarrow \\infty} \\frac{\\sin \\left(constantval / 3^{continuous}\\right)}{constantval / 3^{continuous}}=1,\n\\]\nand it follows that\n\\[\n\\lim _{continuous \\rightarrow \\infty} \\prod_{unindexed=1}^{continuous} \\frac{1+2 \\cos \\left(2 constantval / 3^{unindexed}\\right)}{3}=\\frac{\\sin constantval}{constantval} .\n\\]\n\nRemark. Because of the special behavior of 0 in multiplication, the convergence of infinite products is not usually decided simply by the convergence of the sequence of partial products. Rather, it is required that only finitely many factors be zero and that the partial products of the others have a non-zero limit. (See, for example, Ahlfors, Complex Analysis, 2nd ed., McGraw-Hill, New York, 1966, p. 189.) The infinite product above converges in this restricted sense because for any fixed \\( \\boldsymbol{constantval} \\),\n\\[\n1+2 \\cos \\frac{2 constantval}{3^{unindexed}}\n\\]\ndoes not vanish for large \\( unindexed \\) and the above proof shows that\n\\[\n\\lim _{continuous \\rightarrow \\infty} \\prod_{unindexed=finality+1}^{continuous}\\left(1+2 \\cos \\frac{2 constantval}{3^{unindexed}}\\right)=\\frac{\\sin \\left(constantval / 3^{finality}\\right)}{constantval / 3^{finality}} \\neq 0\n\\]\nfor large \\( finality \\)."
    },
    "garbled_string": {
      "map": {
        "x": "qzxwvtnp",
        "k": "hjgrksla",
        "n": "vbmshqtz",
        "t": "xzcflgpo",
        "\\theta": "mnqrdskj"
      },
      "question": "\\begin{array}{l}\n\\text { 6. Prove that for every real or complex } \\boldsymbol{qzxwvtnp}\\\\\n\\prod_{hjgrksla=1}^{\\infty} \\frac{1+2 \\cos \\frac{2 qzxwvtnp}{3^{hjgrksla}}}{3}=\\frac{\\sin qzxwvtnp}{qzxwvtnp}\n\\end{array}",
      "solution": "Solution. The identity\n\\[\n\\sin 3 mnqrdskj-\\sin mnqrdskj=2 \\sin \\frac{1}{2}(3 mnqrdskj-mnqrdskj) \\cos \\frac{1}{2}(3 mnqrdskj+mnqrdskj)\n\\]\nleads to the identity\n\\[\n\\sin 3 mnqrdskj=\\sin mnqrdskj(1+2 \\cos 2 mnqrdskj) .\n\\]\n\nHence for any \\( \\boldsymbol{qzxwvtnp} \\),\n\\[\n\\begin{aligned}\n\\sin qzxwvtnp & =\\sin (qzxwvtnp / 3)(1+2 \\cos (2 qzxwvtnp / 3)) \\\\\n& =\\sin (qzxwvtnp / 9)(1+2 \\cos (2 qzxwvtnp / 9))(1+2 \\cos (2 qzxwvtnp / 3))\n\\end{aligned}\n\\]\nand after \\( \\boldsymbol{vbmshqtz} \\) iterations we have\n\\[\n\\sin qzxwvtnp=\\sin \\left(qzxwvtnp / 3^{vbmshqtz}\\right) \\prod_{hjgrksla=1}^{vbmshqtz}\\left(1+2 \\cos \\left(2 qzxwvtnp / 3^{hjgrksla}\\right)\\right)\n\\]\n\nFor \\( qzxwvtnp=0 \\), the right member of the required relation is not defined. However, if we interpret \\( (\\sin 0) / 0 \\) as \\( 1\\left(=\\lim _{qzxwvtnp \\rightarrow 0} \\sin qzxwvtnp / qzxwvtnp\\right) \\), then the required equation is correct since every factor in the infinite product is 1.\n\nIf \\( qzxwvtnp \\neq 0 \\), we divide (1) by \\( qzxwvtnp \\) and rearrange to get\n\\[\n\\frac{\\sin qzxwvtnp}{qzxwvtnp}=\\frac{\\sin \\left(qzxwvtnp / 3^{vbmshqtz}\\right)}{qzxwvtnp / 3^{vbmshqtz}} \\prod_{hjgrksla=1}^{vbmshqtz}\\left(\\frac{1+2 \\cos \\left(2 qzxwvtnp / 3^{hjgrksla}\\right)}{3}\\right) .\n\\]\n\nNow\n\\[\n\\lim _{vbmshqtz \\rightarrow \\infty} \\frac{\\sin \\left(qzxwvtnp / 3^{vbmshqtz}\\right)}{qzxwvtnp / 3^{vbmshqtz}}=1,\n\\]\nand it follows that\n\\[\n\\lim _{vbmshqtz \\rightarrow \\infty} \\prod_{hjgrksla=1}^{vbmshqtz} \\frac{1+2 \\cos \\left(2 qzxwvtnp / 3^{hjgrksla}\\right)}{3}=\\frac{\\sin qzxwvtnp}{qzxwvtnp} .\n\\]\n\nRemark. Because of the special behavior of 0 in multiplication, the convergence of infinite products is not usually decided simply by the convergence of the sequence of partial products. Rather, it is required that only finitely many factors be zero and that the partial products of the others have a non-zero limit. (See, for example, Ahlfors, Complex Analysis, 2nd ed., McGraw-Hill, New York, 1966, p. 189.) The infinite product above converges in this restricted sense because for any fixed \\( \\boldsymbol{qzxwvtnp} \\),\n\\[\n1+2 \\cos \\frac{2 qzxwvtnp}{3^{hjgrksla}}\n\\]\ndoes not vanish for large \\( hjgrksla \\) and the above proof shows that\n\\[\n\\lim _{vbmshqtz \\rightarrow \\infty} \\prod_{hjgrksla=xzcflgpo+1}^{vbmshqtz}\\left(1+2 \\cos \\frac{2 qzxwvtnp}{3^{hjgrksla}}\\right)=\\frac{\\sin \\left(qzxwvtnp / 3^{xzcflgpo}\\right)}{qzxwvtnp / 3^{xzcflgpo}} \\neq 0\n\\]\nfor large \\( xzcflgpo \\)."
    },
    "kernel_variant": {
      "question": "Let $m\\ge 2$ be a fixed integer and denote by $U_{m-1}$ the $(m-1)$-st Chebyshev polynomial of the second kind, i.e.  \n\\[\nU_{m-1}(\\cos\\theta)=\\frac{\\sin (m\\theta)}{\\sin\\theta},\\qquad\\theta\\in\\mathbf R .\n\\]\n\n1.  Prove the infinite–product identity  \n\\[\n\\forall z\\in\\mathbf C\\qquad \n\\prod_{k=0}^{\\infty}\\Bigl(\\tfrac1m\\,U_{m-1}\\bigl(\\cos(z/m^{\\,k+1})\\bigr)\\Bigr)=\\frac{\\sin z}{z}.\n\\tag{$\\star$}\n\\]\n\n2.  Show that the product in $(\\star)$ converges absolutely and locally uniformly on $\\mathbf C$; hence the left–hand side defines an entire function.\n\n3.  By taking logarithmic derivatives of the factors in $(\\star)$ and integrating, derive Euler’s classical Weierstrass factorisation  \n\\[\n\\sin z = z\\prod_{n=1}^{\\infty}\\Bigl(1-\\frac{z^{2}}{\\pi^{2}n^{2}}\\Bigr).\n\\tag{$\\heartsuit$}\n\\]\n\n4.  Finally, prove that $\\dfrac{\\sin z}{z}$ is the unique entire function of order $1$ whose non–zero zeros are precisely the integer multiples of $\\pi$ and which satisfies $(\\star)$.  \n(As usual, $\\dfrac{\\sin z}{z}$ is understood at $z=0$ by continuity.)",
      "solution": "Throughout write  \n\\[\nf_m(z):=\\prod_{k=0}^{\\infty}\\Bigl(\\tfrac1m\\,U_{m-1}\\bigl(\\cos(z/m^{\\,k+1})\\bigr)\\Bigr),\\qquad z\\in\\mathbf C .\n\\tag{0}\n\\]\n\nStep 1.  A one–step decomposition of $\\sin$.  \nBecause $U_{m-1}(\\cos\\theta)=\\dfrac{\\sin (m\\theta)}{\\sin\\theta}$ we have  \n\\[\n\\sin (m\\theta)=\\sin\\theta\\;U_{m-1}(\\cos\\theta).\n\\tag{1}\n\\]\nPutting $\\theta=\\dfrac{z}{m}$ gives  \n\\[\n\\sin z=\\sin\\!\\bigl(z/m\\bigr)\\,U_{m-1}\\!\\bigl(\\cos(z/m)\\bigr).\n\\tag{2}\n\\]\n\nStep 2.  Finite iteration.  \nIterating $(2)$ $N$ times yields  \n\\[\n\\sin z=\\sin\\!\\bigl(z/m^{N}\\bigr)\\prod_{k=1}^{N}U_{m-1}\\!\\bigl(\\cos(z/m^{k})\\bigr).\n\\tag{3}\n\\]\nDividing by $z$ and extracting powers of $m$ we obtain  \n\\[\n\\frac{\\sin z}{z}=\\frac{\\sin\\!\\bigl(z/m^{N}\\bigr)}{z/m^{N}}\n              \\prod_{k=1}^{N}\\Bigl(\\tfrac1m\\,U_{m-1}\\bigl(\\cos(z/m^{k})\\bigr)\\Bigr).\n\\tag{4}\n\\]\n\nStep 3.  Absolute and locally uniform convergence (problem 2).  \nFor small $t$  \n\\[\n\\cos t=1-\\frac{t^{2}}{2}+O(t^{4}),\\qquad\nU_{m-1}(\\cos t)=m-\\frac{m(m^{2}-1)}{6}\\,t^{2}+O(t^{4}).\n\\]\nHence  \n\\[\n\\tfrac1m\\,U_{m-1}(\\cos t)=1-\\frac{(m^{2}-1)t^{2}}{6}+O(t^{4}).\n\\tag{5}\n\\]\nPut $t=z/m^{k+1}$.  On the disc $\\lvert z\\rvert\\le R$ one has  \n\\[\n\\bigl\\lvert\\tfrac1m\\,U_{m-1}(\\cos(z/m^{k+1}))-1\\bigr\\rvert\\le C_{R}\\,m^{-2k-2},\n\\]\nso $\\sum_{k\\ge 0}\\lvert\\log\\text{factor}_{k}\\rvert$ converges absolutely and uniformly on $\\lvert z\\rvert\\le R$ (because $\\sum_{k}m^{-2k}<\\infty$).  Therefore the product in $(0)$ converges absolutely and locally uniformly on $\\mathbf C$ and $f_m$ is entire.\n\nStep 4.  Passage to the limit $N\\to\\infty$ (problem 1).  \nSince $\\displaystyle\\lim_{w\\to 0}\\dfrac{\\sin w}{w}=1$, letting $N\\to\\infty$ in $(4)$ and using the uniform convergence just proved we get  \n\\[\nf_m(z)=\\frac{\\sin z}{z},\\qquad z\\in\\mathbf C .\n\\tag{6}\n\\]\n\nStep 5.  Logarithmic derivatives and Euler’s product (problem 3).\n\n5.1  Derivative of a single factor.  \nFor $t=z/m^{k+1}$ set $g(t)=\\log\\bigl(\\dfrac{\\sin(mt)}{m\\sin t}\\bigr)$.  Then\n\\[\ng'(t)=m\\cot(mt)-\\cot t.\n\\]\nTherefore\n\\[\n\\frac{\\mathrm d}{\\mathrm d z}\\log\\!\\Bigl(\\tfrac1m\\,U_{m-1}(\\cos(z/m^{k+1}))\\Bigr)\n          =\\frac1{m^{k}}\\cot\\!\\bigl(z/m^{k}\\bigr)\n           -\\frac1{m^{k+1}}\\cot\\!\\bigl(z/m^{k+1}\\bigr).\n\\tag{7}\n\\]\n\n5.2  Telescoping in $k$.  \nSumming $(7)$ from $k=0$ to $N$ we obtain  \n\\[\nS_N(z):=\\sum_{k=0}^{N}\\frac{\\mathrm d}{\\mathrm dz}\\log\\!\\Bigl(\\tfrac1m\\,U_{m-1}(\\cos(z/m^{k+1}))\\Bigr)\n      =\\cot z-\\frac1{m^{N+1}}\\cot\\!\\Bigl(\\tfrac{z}{m^{N+1}}\\Bigr).\n\\tag{8}\n\\]\nAs $w\\to 0$, $\\cot w=\\dfrac1w-\\dfrac w3+O(w^{3})$, whence  \n\\[\n\\frac1{m^{N+1}}\\cot\\!\\Bigl(\\tfrac{z}{m^{N+1}}\\Bigr)\n      =\\frac1z-\\frac{z}{3m^{2N+2}}+O(m^{-4N-4})\\xrightarrow[N\\to\\infty]{}\\frac1z .\n\\]\nHence\n\\[\n\\lim_{N\\to\\infty}S_N(z)=\\cot z-\\frac1z .\n\\tag{9}\n\\]\n\n5.3  Agreement with the right–hand side.  \nBecause of $(6)$, $f_m(z)=\\dfrac{\\sin z}{z}$, whose logarithmic derivative is precisely $\\cot z-\\dfrac1z$.  The limit in $(9)$ therefore equals $\\dfrac{\\mathrm d}{\\mathrm dz}\\log f_m(z)$, as required.\n\n5.4  Partial–fraction expansion.  \nThe meromorphic function $\\cot z-\\dfrac1z$ has simple poles at $z=\\pm\\pi n$ $(n\\in\\mathbf Z\\setminus\\{0\\})$ with residue $+1$.  Its classical expansion is  \n\\[\n\\cot z-\\frac1z=\\sum_{n=1}^{\\infty}\\frac{2z}{z^{2}-\\pi^{2}n^{2}},\\qquad z\\notin\\pi\\mathbf Z,\n\\tag{10}\n\\]\nwhere the series converges absolutely and locally uniformly because $\\sum_{n\\ge 1}n^{-2}<\\infty$.\n\n5.5  Integration.  \nIntegrating $(10)$ from $0$ to $z$ along any path avoiding $\\pi\\mathbf Z$ we get  \n\\[\n\\log(\\sin z)-\\log z=\\sum_{n=1}^{\\infty}\\log\\Bigl(1-\\frac{z^{2}}{\\pi^{2}n^{2}}\\Bigr).\n\\]\nSince both sides vanish at $z=0$, no additive constant appears.  Exponentiating yields Euler’s factorisation $(\\heartsuit)$.\n\nStep 6.  Uniqueness (problem 4).  \nLet $g$ be an entire function of order $1$ whose non–zero zeros are the simple zeros $\\pi n$, $n\\in\\mathbf Z\\setminus\\{0\\}$, and which satisfies $(\\star)$.  Then $g$ and $\\dfrac{\\sin z}{z}$ share the same zeros and the same canonical product (the left–hand side of $(\\star)$).  Consequently\n\\[\nh(z):=\\frac{g(z)}{\\sin z/z}\n\\]\nis an entire function without zeros.  Moreover $h(0)=1$ (limit value) and $h$ is of order at most $1$.  By standard Hadamard factorisation, an entire function of finite order with no zeros must be of the form $e^{az+b}$; the condition $h(0)=1$ forces $b=0$.  Inserting this into $(\\star)$ we get $h(z)\\equiv 1$ (otherwise $(\\star)$ fails at $z=0$).  Hence $g(z)=\\dfrac{\\sin z}{z}$, completing the proof of uniqueness.\n\n\\[\n\\boxed{\\;f_m(z)=\\dfrac{\\sin z}{z}\\;}\n\\qquad\\text{for all }z\\in\\mathbf C.\\quad\\blacksquare\n\\]",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.425450",
        "was_fixed": false,
        "difficulty_analysis": "• Higher algebraic structure: the problem replaces the elementary cosine factor by a Chebyshev polynomial Uₘ₋₁, demanding familiarity with special polynomials and the identity sin(m θ)=sin θ Uₘ₋₁(cos θ).\n\n• Additional parameters: the integer m ≥ 2 introduces a family of identities; handling an arbitrary m requires uniform estimates and careful parameter tracking.\n\n• Deeper analytic requirements: the proof now needs absolute and locally uniform convergence of an infinite product of non-trivial analytic factors, invoking Taylor expansions and uniform majorants.\n\n• Entire-function theory: deriving the Weierstrass product and proving uniqueness obliges the solver to use Hadamard’s factorisation theorem and growth–order arguments—well beyond the techniques needed for the original cosine product.\n\n• Multi-layer reasoning: the solution combines trigonometric identities, polynomial theory, complex analysis of infinite products, and entire-function uniqueness, providing substantially more conceptual and technical depth than either the original problem or the simpler “cos(z/2^{k})” kernel variant."
      }
    },
    "original_kernel_variant": {
      "question": "Let $m\\ge 2$ be a fixed integer and denote by $U_{m-1}$ the $(m-1)$-st Chebyshev polynomial of the second kind, i.e.  \n\\[\nU_{m-1}(\\cos\\theta)=\\frac{\\sin (m\\theta)}{\\sin\\theta},\\qquad\\theta\\in\\mathbf R .\n\\]\n\n1.  Prove the infinite–product identity  \n\\[\n\\forall z\\in\\mathbf C\\qquad \n\\prod_{k=0}^{\\infty}\\Bigl(\\tfrac1m\\,U_{m-1}\\bigl(\\cos(z/m^{\\,k+1})\\bigr)\\Bigr)=\\frac{\\sin z}{z}.\n\\tag{$\\star$}\n\\]\n\n2.  Show that the product in $(\\star)$ converges absolutely and locally uniformly on $\\mathbf C$; hence the left–hand side defines an entire function.\n\n3.  By taking logarithmic derivatives of the factors in $(\\star)$ and integrating, derive Euler’s classical Weierstrass factorisation  \n\\[\n\\sin z = z\\prod_{n=1}^{\\infty}\\Bigl(1-\\frac{z^{2}}{\\pi^{2}n^{2}}\\Bigr).\n\\tag{$\\heartsuit$}\n\\]\n\n4.  Finally, prove that $\\dfrac{\\sin z}{z}$ is the unique entire function of order $1$ whose non–zero zeros are precisely the integer multiples of $\\pi$ and which satisfies $(\\star)$.  \n(As usual, $\\dfrac{\\sin z}{z}$ is understood at $z=0$ by continuity.)",
      "solution": "Throughout write  \n\\[\nf_m(z):=\\prod_{k=0}^{\\infty}\\Bigl(\\tfrac1m\\,U_{m-1}\\bigl(\\cos(z/m^{\\,k+1})\\bigr)\\Bigr),\\qquad z\\in\\mathbf C .\n\\tag{0}\n\\]\n\nStep 1.  A one–step decomposition of $\\sin$.  \nBecause $U_{m-1}(\\cos\\theta)=\\dfrac{\\sin (m\\theta)}{\\sin\\theta}$ we have  \n\\[\n\\sin (m\\theta)=\\sin\\theta\\;U_{m-1}(\\cos\\theta).\n\\tag{1}\n\\]\nPutting $\\theta=\\dfrac{z}{m}$ gives  \n\\[\n\\sin z=\\sin\\!\\bigl(z/m\\bigr)\\,U_{m-1}\\!\\bigl(\\cos(z/m)\\bigr).\n\\tag{2}\n\\]\n\nStep 2.  Finite iteration.  \nIterating $(2)$ $N$ times yields  \n\\[\n\\sin z=\\sin\\!\\bigl(z/m^{N}\\bigr)\\prod_{k=1}^{N}U_{m-1}\\!\\bigl(\\cos(z/m^{k})\\bigr).\n\\tag{3}\n\\]\nDividing by $z$ and extracting powers of $m$ we obtain  \n\\[\n\\frac{\\sin z}{z}=\\frac{\\sin\\!\\bigl(z/m^{N}\\bigr)}{z/m^{N}}\n              \\prod_{k=1}^{N}\\Bigl(\\tfrac1m\\,U_{m-1}\\bigl(\\cos(z/m^{k})\\bigr)\\Bigr).\n\\tag{4}\n\\]\n\nStep 3.  Absolute and locally uniform convergence (problem 2).  \nFor small $t$  \n\\[\n\\cos t=1-\\frac{t^{2}}{2}+O(t^{4}),\\qquad\nU_{m-1}(\\cos t)=m-\\frac{m(m^{2}-1)}{6}\\,t^{2}+O(t^{4}).\n\\]\nHence  \n\\[\n\\tfrac1m\\,U_{m-1}(\\cos t)=1-\\frac{(m^{2}-1)t^{2}}{6}+O(t^{4}).\n\\tag{5}\n\\]\nPut $t=z/m^{k+1}$.  On the disc $\\lvert z\\rvert\\le R$ one has  \n\\[\n\\bigl\\lvert\\tfrac1m\\,U_{m-1}(\\cos(z/m^{k+1}))-1\\bigr\\rvert\\le C_{R}\\,m^{-2k-2},\n\\]\nso $\\sum_{k\\ge 0}\\lvert\\log\\text{factor}_{k}\\rvert$ converges absolutely and uniformly on $\\lvert z\\rvert\\le R$ (because $\\sum_{k}m^{-2k}<\\infty$).  Therefore the product in $(0)$ converges absolutely and locally uniformly on $\\mathbf C$ and $f_m$ is entire.\n\nStep 4.  Passage to the limit $N\\to\\infty$ (problem 1).  \nSince $\\displaystyle\\lim_{w\\to 0}\\dfrac{\\sin w}{w}=1$, letting $N\\to\\infty$ in $(4)$ and using the uniform convergence just proved we get  \n\\[\nf_m(z)=\\frac{\\sin z}{z},\\qquad z\\in\\mathbf C .\n\\tag{6}\n\\]\n\nStep 5.  Logarithmic derivatives and Euler’s product (problem 3).\n\n5.1  Derivative of a single factor.  \nFor $t=z/m^{k+1}$ set $g(t)=\\log\\bigl(\\dfrac{\\sin(mt)}{m\\sin t}\\bigr)$.  Then\n\\[\ng'(t)=m\\cot(mt)-\\cot t.\n\\]\nTherefore\n\\[\n\\frac{\\mathrm d}{\\mathrm d z}\\log\\!\\Bigl(\\tfrac1m\\,U_{m-1}(\\cos(z/m^{k+1}))\\Bigr)\n          =\\frac1{m^{k}}\\cot\\!\\bigl(z/m^{k}\\bigr)\n           -\\frac1{m^{k+1}}\\cot\\!\\bigl(z/m^{k+1}\\bigr).\n\\tag{7}\n\\]\n\n5.2  Telescoping in $k$.  \nSumming $(7)$ from $k=0$ to $N$ we obtain  \n\\[\nS_N(z):=\\sum_{k=0}^{N}\\frac{\\mathrm d}{\\mathrm dz}\\log\\!\\Bigl(\\tfrac1m\\,U_{m-1}(\\cos(z/m^{k+1}))\\Bigr)\n      =\\cot z-\\frac1{m^{N+1}}\\cot\\!\\Bigl(\\tfrac{z}{m^{N+1}}\\Bigr).\n\\tag{8}\n\\]\nAs $w\\to 0$, $\\cot w=\\dfrac1w-\\dfrac w3+O(w^{3})$, whence  \n\\[\n\\frac1{m^{N+1}}\\cot\\!\\Bigl(\\tfrac{z}{m^{N+1}}\\Bigr)\n      =\\frac1z-\\frac{z}{3m^{2N+2}}+O(m^{-4N-4})\\xrightarrow[N\\to\\infty]{}\\frac1z .\n\\]\nHence\n\\[\n\\lim_{N\\to\\infty}S_N(z)=\\cot z-\\frac1z .\n\\tag{9}\n\\]\n\n5.3  Agreement with the right–hand side.  \nBecause of $(6)$, $f_m(z)=\\dfrac{\\sin z}{z}$, whose logarithmic derivative is precisely $\\cot z-\\dfrac1z$.  The limit in $(9)$ therefore equals $\\dfrac{\\mathrm d}{\\mathrm dz}\\log f_m(z)$, as required.\n\n5.4  Partial–fraction expansion.  \nThe meromorphic function $\\cot z-\\dfrac1z$ has simple poles at $z=\\pm\\pi n$ $(n\\in\\mathbf Z\\setminus\\{0\\})$ with residue $+1$.  Its classical expansion is  \n\\[\n\\cot z-\\frac1z=\\sum_{n=1}^{\\infty}\\frac{2z}{z^{2}-\\pi^{2}n^{2}},\\qquad z\\notin\\pi\\mathbf Z,\n\\tag{10}\n\\]\nwhere the series converges absolutely and locally uniformly because $\\sum_{n\\ge 1}n^{-2}<\\infty$.\n\n5.5  Integration.  \nIntegrating $(10)$ from $0$ to $z$ along any path avoiding $\\pi\\mathbf Z$ we get  \n\\[\n\\log(\\sin z)-\\log z=\\sum_{n=1}^{\\infty}\\log\\Bigl(1-\\frac{z^{2}}{\\pi^{2}n^{2}}\\Bigr).\n\\]\nSince both sides vanish at $z=0$, no additive constant appears.  Exponentiating yields Euler’s factorisation $(\\heartsuit)$.\n\nStep 6.  Uniqueness (problem 4).  \nLet $g$ be an entire function of order $1$ whose non–zero zeros are the simple zeros $\\pi n$, $n\\in\\mathbf Z\\setminus\\{0\\}$, and which satisfies $(\\star)$.  Then $g$ and $\\dfrac{\\sin z}{z}$ share the same zeros and the same canonical product (the left–hand side of $(\\star)$).  Consequently\n\\[\nh(z):=\\frac{g(z)}{\\sin z/z}\n\\]\nis an entire function without zeros.  Moreover $h(0)=1$ (limit value) and $h$ is of order at most $1$.  By standard Hadamard factorisation, an entire function of finite order with no zeros must be of the form $e^{az+b}$; the condition $h(0)=1$ forces $b=0$.  Inserting this into $(\\star)$ we get $h(z)\\equiv 1$ (otherwise $(\\star)$ fails at $z=0$).  Hence $g(z)=\\dfrac{\\sin z}{z}$, completing the proof of uniqueness.\n\n\\[\n\\boxed{\\;f_m(z)=\\dfrac{\\sin z}{z}\\;}\n\\qquad\\text{for all }z\\in\\mathbf C.\\quad\\blacksquare\n\\]",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.369539",
        "was_fixed": false,
        "difficulty_analysis": "• Higher algebraic structure: the problem replaces the elementary cosine factor by a Chebyshev polynomial Uₘ₋₁, demanding familiarity with special polynomials and the identity sin(m θ)=sin θ Uₘ₋₁(cos θ).\n\n• Additional parameters: the integer m ≥ 2 introduces a family of identities; handling an arbitrary m requires uniform estimates and careful parameter tracking.\n\n• Deeper analytic requirements: the proof now needs absolute and locally uniform convergence of an infinite product of non-trivial analytic factors, invoking Taylor expansions and uniform majorants.\n\n• Entire-function theory: deriving the Weierstrass product and proving uniqueness obliges the solver to use Hadamard’s factorisation theorem and growth–order arguments—well beyond the techniques needed for the original cosine product.\n\n• Multi-layer reasoning: the solution combines trigonometric identities, polynomial theory, complex analysis of infinite products, and entire-function uniqueness, providing substantially more conceptual and technical depth than either the original problem or the simpler “cos(z/2^{k})” kernel variant."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}