Odd function means that any x in the definition domain of the function f(x) whose definition domain is symmetrical about the origin has f(-x)=-f(x), so the function f(x) is called odd function.
1727, Euler, a young Swiss mathematician, first put forward the concept of parity function in a paper (Latin) submitted to St. Petersburg Academy of Sciences to solve the "rebound path problem".
nature
1, the difference between the sum or subtraction of two odd function is odd function.
2. The difference between the sum or subtraction of even and odd functions is the parity function.
3. The quotient obtained by multiplication or division of two odd function is an even function.
4. The quotient of the product or division of an even function multiplied by a odd function is odd function.
5. It is both a odd function and an even function. Odd function's integral on the symmetric interval is zero.
Euler's earliest definition
If x is replaced by -x and the function remains the same, such a function is called an even function. Euler enumerated three types of even functions and three types of odd function, and discussed the properties of even functions and odd function. The French mathematician D'Alembert edited Volume 7 of the Encyclopedia of Diderot (published in 1757).
The entry on function says: "Ancient geometricians, or more accurately, ancient analysts, called the different powers of a certain quantity x a function of X." Similarly, French mathematician Lagrange said in the opening of analytic function theory (1797) that early analysts only used the word "function" to mean "different powers of the same quantity". Later,
Leibniz and johann bernoulli first adopted the latter meaning, that is, "all quantities derived from other quantities in any way". In the paper of 1727, Euler did not involve any transcendental function when discussing the odd and even functions. Therefore, the earliest concepts of even and odd functions are all aimed at power functions and related composite functions.
The names of "odd function" and "even function" put forward by Euler are obviously derived from the exponents or parity of power functions: power functions with even exponents are even functions, and power functions with odd exponents are odd function.