How do you know when an expression is not solvable?

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In summary, the conversation discusses how to determine if a mathematical expression, such as an integral or differential equation, is solvable or not. The speaker mentions that it is usually difficult to determine this and references Galois theory and differential Galois theory as methods for proving solvability. They also mention a PDF that provides more information on this topic.
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Repetit
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My question pretty much is the title of this post, but let me explain it a bit more. When in physics or mathematics, or some other discipline involving math, you run into an integral, a differential equation or some other expression, how do you know if it's solvable or not? I mean, how do I know when to tend to numerical computations because an analytical result cannot be obtained? How do I know that my incapability to solve a certain equation is not just due to my lack of knowledge, but because the equation is, in fact, now solvable?

For example, how would I know that the integral:

[tex]
\int e^{-x^2} dx
[/tex]

is not solvable?

Thanks in advance
 
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  • #2
I don't think there is anyway of knowing whether an integral is possible or not unless it's been proven one way or another for that specific integral.
 
  • #3
It's usually very difficult. You know how Galois theory helps us determine the solvability of polynomial equations? Well, there is a field of math known as differential Galois theory that can be used to prove that the integral you quoted (for instance) is not solvable, in the sense that it cannot be expressed in terms of elementary functions. Although personally I'm ignorant of its ways!

You can also read the following pdf (which AFAIK was posted on these forums):
http://www.claymath.org/programs/outreach/academy/LectureNotes05/Conrad.pdf [Broken]
 
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1. What is an expression?

An expression is a mathematical statement that contains variables, numbers, and mathematical operations such as addition, subtraction, multiplication, and division.

2. How do you determine if an expression is solvable?

An expression is solvable if it can be simplified or evaluated to a specific numerical value. This can be done by following the order of operations, combining like terms, and solving for any variables.

3. What makes an expression not solvable?

An expression is not solvable if it contains undefined or imaginary numbers, such as dividing by zero or taking the square root of a negative number. It can also be not solvable if there are too many variables or if the expression cannot be simplified further.

4. Are there any rules for determining if an expression is not solvable?

Yes, there are rules and properties in mathematics that can help determine if an expression is not solvable. Some examples include the associative and commutative properties, the distributive property, and the rules for exponents and logarithms.

5. Can an expression be partially solvable?

Yes, an expression can be partially solvable if some parts of it can be simplified or evaluated, while other parts are not solvable. This can happen when there are multiple variables or different mathematical operations involved in the expression.

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