ඉරට්ටේ හා ඔත්තේ ශ්‍රිත

In mathematics, ඉරට්ටේ ශ්‍රිත හා ඔත්තේ ශ්‍රිත are which satisfy particular relations, with respect to taking . They are important in many areas of mathematical analysis, especially the theory of and . They are named for the of the powers of the which satisfy each condition: the function f(x) = xⁿ is an even function if n is an even integer, and it is an odd function if n is an odd integer.

ඉරට්ටේ ශ්‍රිත

ƒ(x) = x² ඉරට්ටේ ශ්‍රිතයක් සඳහා උදාහරණයකි.

ƒ(x) = x³ ඔත්තේ ශ්‍රිතයක් සඳහා උදාහරණයකි.

ƒ(x) = x³ + 1 ඔත්තේ හෝ ඉරට්ටේ දෙකම නොවන.

Let f(x) be a real-valued function of a real variable. Then f is even if the following equation holds for all x in the domain of f:

${\displaystyle f(x)=f(-x).\,}$

ජ්‍යාමිතිකව , ඉරට්ටෙ ශ්‍රිතයක ප්‍රස්තාරය with respect to the y-axis, meaning that its remains unchanged after about the y-axis.

ඉරට්ටෙ ශ්‍රිත සඳහා උදාහරණ ලෙස , x², x⁴, (x), and (x).

ඔත්තේ ශ්‍රිත

Again, let f(x) be a තාත්වික-valued function of a real variable. Then f is odd if the following equation holds for all x in the domain of f:

${\displaystyle -f(x)=f(-x)\,,}$

or

${\displaystyle f(x)+f(-x)=0\,.}$

Geometrically, the graph of an odd function has rotational symmetry with respect to the , meaning that its remains unchanged after of 180 about the origin.

Examples of odd functions are x, x³, (x), sinh(x), and (x).

සමහර තොරතුරු

Note: A function's being odd or even does not imply differentiability, or even continuity. For example, the is even, but is nowhere continuous. Properties involving Fourier series, Taylor series, derivatives and so on may only be used when they can be assumed to exist.

මූලික ගුණ

• The only function which is both even and odd is the which is identically zero (i.e., f(x) = 0 for all x).
• The of an even and odd function is neither even nor odd, unless one of the functions is identically zero.
• The sum of two even functions is even, and any constant multiple of an even function is even.
• The sum of two odd functions is odd, and any constant multiple of an odd function is odd.
• The of two even functions is an even function.
• The product of two odd functions is an even function.
• The product of an even function and an odd function is an odd function.
• The of two even functions is an even function.
• The quotient of two odd functions is an even function.
• The quotient of an even function and an odd function is an odd function.
• The of an even function is odd.
• The derivative of an odd function is even.
• The of two even functions is even, and the composition of two odd functions is odd.
• The composition of an even function and an odd function is even.
• The composition of any function with an even function is even (but not vice versa).
• The of an odd function from −A to +A is zero (where A is finite, and the function has no vertical asymptotes between −A and A).
• The integral of an even function from −A to +A is twice the integral from 0 to +A (where A is finite, and the function has no vertical asymptotes between −A and A).

ශ්‍රේණි

• The of an even function includes only even powers.
• The Maclaurin series of an odd function includes only odd powers.
• The of a even function includes only terms.
• The Fourier series of a periodic odd function includes only terms.

වීජීය වියුහය

• Any of even functions is even, and the even functions form a over the reals. Similarly, any linear combination of odd functions is odd, and the odd functions also form a vector space over the reals. In fact, the vector space of all real-valued functions is the of the of even and odd functions. In other words, every function f(x) can be written uniquely as the sum of an even function and an odd function:

${\displaystyle f(x)=f_{\text{even}}(x)+f_{\text{odd}}(x)\,,}$

where

${\displaystyle f_{\text{even}}(x)={\frac {f(x)+f(-x)}{2}}}$

is even and

${\displaystyle f_{\text{odd}}(x)={\frac {f(x)-f(-x)}{2}}}$

is odd.

• The even functions form a over the reals. However, the odd functions do not form an algebra over the reals.

ප්‍රසංවාද

In , occurs when a signal is multiplied by a non-linear . The type of produced depend on the transfer function:

• When the transfer function is even, the resulting signal will consist of only even harmonics of the input sine wave; ${\displaystyle 2f,4f,6f,\dots \ }$
  • The is also an odd harmonic, so will not be present.
  • A simple example is a .
• When it is odd, the resulting signal will consist of only odd harmonics of the input sine wave; ${\displaystyle 1f,3f,5f,\dots \ }$
  • The output signal will be half-wave .
  • A simple example is in a symmetric .
• When it is asymmetric, the resulting signal may contain either even or odd harmonics; ${\displaystyle 1f,2f,3f,\dots \ }$
  • A simple example is clipping in an asymmetrical .

යොමුව

1. Ask the Doctors: Tube vs. Solid-State Harmonics

තවදුරටත්

• සංකීර්ණ සංඛ්‍යා වල සාමාන්‍යයීකරණය සඳහා.
• ටේලර් ශ්‍රේණිය
• ෆෝරියර් ශ්‍රේණිය