Why there's no solution of closed form for the transcendental equations?












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I just saw this question and I saw this kind of question in MSE a lot before. Also, the answers are always same: such equations can't be solved as a closed form in general, and we can only find its numerical solution. I also believe this, but why? There are some similar questions in integral, and I know that can be answered by Liouville's theorem, which proves that integrals such as
$$
int e^{x^{2}}dx, quad int frac{sin x}{x} dx, quad int sqrt{1-x^{4}}dx, quad intfrac{1}{log x}dx
$$

can't be solved in terms of elementary functions (that can be expressed as polynomials, trigonometry functions, exponential functions, logarithmic functions, and their composition and $n$-th root).
Do we have a similar theorem for equations such as $e^{x}(x^{2}+3x+1) = 5sin (log x)$? Thanks in advance.










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    0














    I just saw this question and I saw this kind of question in MSE a lot before. Also, the answers are always same: such equations can't be solved as a closed form in general, and we can only find its numerical solution. I also believe this, but why? There are some similar questions in integral, and I know that can be answered by Liouville's theorem, which proves that integrals such as
    $$
    int e^{x^{2}}dx, quad int frac{sin x}{x} dx, quad int sqrt{1-x^{4}}dx, quad intfrac{1}{log x}dx
    $$

    can't be solved in terms of elementary functions (that can be expressed as polynomials, trigonometry functions, exponential functions, logarithmic functions, and their composition and $n$-th root).
    Do we have a similar theorem for equations such as $e^{x}(x^{2}+3x+1) = 5sin (log x)$? Thanks in advance.










    share|cite|improve this question

























      0












      0








      0







      I just saw this question and I saw this kind of question in MSE a lot before. Also, the answers are always same: such equations can't be solved as a closed form in general, and we can only find its numerical solution. I also believe this, but why? There are some similar questions in integral, and I know that can be answered by Liouville's theorem, which proves that integrals such as
      $$
      int e^{x^{2}}dx, quad int frac{sin x}{x} dx, quad int sqrt{1-x^{4}}dx, quad intfrac{1}{log x}dx
      $$

      can't be solved in terms of elementary functions (that can be expressed as polynomials, trigonometry functions, exponential functions, logarithmic functions, and their composition and $n$-th root).
      Do we have a similar theorem for equations such as $e^{x}(x^{2}+3x+1) = 5sin (log x)$? Thanks in advance.










      share|cite|improve this question













      I just saw this question and I saw this kind of question in MSE a lot before. Also, the answers are always same: such equations can't be solved as a closed form in general, and we can only find its numerical solution. I also believe this, but why? There are some similar questions in integral, and I know that can be answered by Liouville's theorem, which proves that integrals such as
      $$
      int e^{x^{2}}dx, quad int frac{sin x}{x} dx, quad int sqrt{1-x^{4}}dx, quad intfrac{1}{log x}dx
      $$

      can't be solved in terms of elementary functions (that can be expressed as polynomials, trigonometry functions, exponential functions, logarithmic functions, and their composition and $n$-th root).
      Do we have a similar theorem for equations such as $e^{x}(x^{2}+3x+1) = 5sin (log x)$? Thanks in advance.







      abstract-algebra






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      asked Nov 24 at 3:09









      Seewoo Lee

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