Why there's no solution of closed form for the transcendental equations?
$begingroup$
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
asked Nov 24 '18 at 3:09
Seewoo LeeSeewoo Lee
6,267826
$endgroup$
add a comment |
$begingroup$
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
asked Nov 24 '18 at 3:09
Seewoo LeeSeewoo Lee
6,267826
$endgroup$
add a comment |
$begingroup$
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
asked Nov 24 '18 at 3:09
Seewoo LeeSeewoo Lee
6,267826
$endgroup$
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
abstract-algebra
asked Nov 24 '18 at 3:09
Seewoo LeeSeewoo Lee
6,267826
asked Nov 24 '18 at 3:09
Seewoo LeeSeewoo Lee
6,267826
asked Nov 24 '18 at 3:09
Seewoo LeeSeewoo Lee
6,267826
asked Nov 24 '18 at 3:09
Seewoo LeeSeewoo Lee
6,267826
asked Nov 24 '18 at 3:09
Seewoo LeeSeewoo Lee
6,267826
6,267826
add a comment |
add a comment |
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