Symmetry When calculating the double integral, it depends on whether the integrand or a part of the integrand is symmetrical about a certain coordinate and whether the integral interval is symmetrical. If possible, you can use symmetry to multiply the integral by 2.

Leibniz and johann bernoulli first adopted the latter meaning. In the paper of 1727, Euler did not involve any transcendental function when discussing the odd and even functions.

Therefore, the earliest concept of parity function is aimed at power function and related composite function. The names of "odd function" and "even function" put forward by Euler are obviously derived from the parity of exponent or exponent molecule of power function: power function with even exponent is even function, and power function with odd exponent is odd function.

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 encyclopedia edited by the French mathematician J.R.D Alembert (17 17- 1783) in Diderot (17 13- 1784).