Mathematics
Integrating Even and Odd Functions
Integrating even and odd functions involves finding the definite integral of a function over a symmetric interval. For even functions, the integral over a symmetric interval is twice the integral over half the interval. Odd functions have a zero integral over a symmetric interval. This concept is useful for simplifying integration problems and understanding the properties of functions.
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4 Key excerpts on "Integrating Even and Odd Functions"
eBook - ePubEngineering Mathematics
A Programmed Approach, 3th Edition
- C W. Evans, C. Evans(Authors)
- 2019(Publication Date)
- CRC Press(Publisher)
21.1). Fig. 21.1: An odd function Suppose f (x) is defined on the interval − π ⩽ x ⩽ π, then f is said to be an even function if f (− x) = f (x) whenever − π ≤ x ≤ π Here are a few examples of even functions: l, x 2, cos x, cosh x. Even functions are easily recognized by their graphs; they are symmetrical about the y -axis (Fig. 21.2). Fig. 21.2: An even function In fact the identity f (x) ≡ [ f (x) + f (− x) 2 ] + [ f (x) − f (− x) 2 ] shows that every function f defined on the interval (− π, π) can be regarded as the sum of an even function and an odd function. In the previous example we found a trigonometrical series corresponding to an odd function and it had one curious feature. Did you notice what it was? There were no cosine terms whatever. We went through the motions of calculating the Fourier coefficients a n only to find they were all zero. Was this a coincidence or is there something deeper here? Let us suppose that f (x) = a 0 2 + ∑ r = 1 ∞ { a r cos r x + b r sin r x } and that f is an odd function on the interval (− π, π). We have already derived formulae for the Fourier coefficients and we. have a n = 1 π ∫ − π π f (x) cos n x d x b n = 1 π ∫ − π π f (x) sin n x d x Therefore a n = 1 π ∫ − π π f (x) cos n x d x = 1 π { ∫ − π 0 f (x) cos n x d x + ∫ 0 π[--=PLGO-SEPARATO. R=--]f (x) cos n x d x } Putting t = −x in the first integral and using f (− t) = − f (t) we. obtain a n = 1 π { ∫ π 0 f (− t) cos (− n t) (− d t) + ∫ 0 π f (x) cos n x d x } = 1 π { ∫ π 0 f (t) cos n t d t + ∫ 0 π f (x) cos n x d x } = 1 π { ∫ 0 π f. (t) cos n t d t + ∫ 0 π f (x) cos n x d x } Observe that t is a dummy variable in the first integral just as x is a dummy variable in the second. Consequently these two integrals are equal and so cancel one another out. Therefore we have shown that for an odd function a n = 0. Moreover we can use the same idea to simplify slightly the formula for b n in the case of an odd function
- William E. Boyce, Richard C. DiPrima, Douglas B. Meade(Authors)
- 2017(Publication Date)
- Wiley(Publisher)
Note that according to equation (2), f (0) must be zero if f is an odd function whose domain contains the origin. Most functions are neither even nor odd; an example is e x . Only one function, f identically zero, is both even and odd. y x (a) y x (b) FIGURE 10.4.1 ( a) An even function. ( b) An odd function. Elementary properties of even and odd functions include the following: 8 1. The sum (difference) and product (quotient) of two even functions are even. 2. The sum (difference) of two odd functions is odd; the product (quotient) of two odd functions is even. ......................................................................................................................................................................... 8 These statements may need to be modified if either function vanishes identically. 10.4 Even and Odd Functions 483 3. The sum (difference) of an odd function and an even function is neither even nor odd; the product (quotient) of an odd function and an even function is odd. The proofs of all these assertions are simple and follow directly from the definitions. For example, if both f 1 and f 2 are odd, and if g( x ) = f 1 ( x ) + f 2 ( x ), then g( −x ) = f 1 ( −x ) + f 2 ( −x ) = − f 1 ( x ) − f 2 ( x ) = − ( f 1 ( x ) + f 2 ( x ) ) = −g( x ), (3) so f 1 + f 2 is an odd function also. Similarly, if h ( x ) = f 1 ( x ) f 2 ( x ), then h ( −x ) = f 1 ( −x ) f 2 ( −x ) = ( − f 1 ( x ) )( − f 2 ( x ) ) = f 1 ( x ) f 2 ( x ) = h ( x ), (4) so that f 1 f 2 is even. Also of importance are the following two integral properties of even and odd functions: 4. If f is an even function, then L −L f ( x ) dx = 2 L 0 f ( x ) dx . (5) 5. If f is an odd function, then L −L f ( x ) dx = 0. (6) Properties 4 and 5 are intuitively clear from the interpretation of an integral in terms of area under a curve, and they also follow immediately from the definitions. For example, if f is even, then L −L f ( x ) dx = 0 −L f ( x ) dx + L 0 f ( x ) dx .
eBook - PDF- Dave Benson(Author)
- 2006(Publication Date)
- Cambridge University Press(Publisher)
Given any function f (θ ), we can obtain an even function by taking the average of f (θ ) and f (−θ ), i.e., 1 2 ( f (θ ) + f (−θ )). Similarly, 1 2 ( f (θ ) − f (−θ )) is an odd function. These add up to give the original function f (θ ), so we have written f (θ ) as a sum of its even part and its odd part, f (θ ) = f (θ ) + f (−θ ) 2 + f (θ ) − f (−θ ) 2 . To see that this is the unique way to write the function as a sum of an even function and an odd function, let us suppose that we are given two expressions f (θ ) = g 1 (θ ) + h 1 (θ ) and f (θ ) = g 2 (θ ) + h 2 (θ ) with g 1 and g 2 even, and h 1 and h 2 odd. Rearranging g 1 + h 1 = g 2 + h 2 , we get g 1 − g 2 = h 2 − h 1 . The left side is even and the right side is odd, so their common value is both even and odd, and hence zero. This means that g 1 = g 2 and h 1 = h 2 . Multiplication of even and odd functions works like addition (and not multipli- cation) of even and odd numbers: × even odd even even odd odd odd even Now for any odd function f (θ ), and for any a > 0, we have 0 −a f (θ ) dθ = − a 0 f (θ ) dθ 2.3 Even and odd functions 45 so that a −a f (θ ) dθ = 0. So, for example, if f (θ ) is even and periodic with period 2π , then sin(mθ ) f (θ ) is odd, and so the Fourier coefficients b m are zero, since b m = 1 π 2π 0 sin(mθ ) f (θ ) dθ = 1 π π −π sin(mθ ) f (θ ) dθ = 0. Similarly, if f (θ ) is odd and periodic with period 2π , then cos(mθ ) f (θ ) is odd, and so the Fourier coefficients a m are zero, since a m = 1 π 2π 0 cos(mθ ) f (θ ) dθ = 1 π π −π cos(mθ ) f (θ ) dθ = 0. This explains, for example, why a m = 0 in the example on page 41. The square wave is not quite an even function, because f (π ) = f (−π ), but changing the value of a function at a finite set of points in the interval of integration never affects the value of an integral, so we just replace f (π ) and f (−π ) by zero to obtain an even function with the same Fourier coefficients.
- Anthony Orton(Author)
- 2004(Publication Date)
- Continuum(Publisher)
The first part of the definition of Figure 3.2 Partitioning 8 into two equal parts an odd number stands in its own right independent of that of an even number. The second, alternative, definition relates an odd number to an even number, overlooking the fact that an odd number differs by 1 from two even numbers, just as every even number differs by 1 from two odd numbers. To many this would not be considered a definition, but a relationship that is the outcome of the definitions of even and odd numbers. Although odd and even numbers are found in most curricula, seldom are the numbers explored to unearth their true richness. Eagle (1995, p. 23) suggests that there are many propositions about odd and even numbers that Euclid listed in Book IX of the 'Elements' 'which are quite accessible' to children. 'Pupils can investigate for themselves what rules exist and try to justify them'. She lists just a few: If you add together as many even numbers as you like, the total is even. If you subtract an odd number from an even number ... If you multiply an odd number by an odd number ... If an odd number divides exactly into an even number, it will also divide exactly into half of it. or 34 Pattern in the Teaching and Learning of Maths Even and odd numbers recur throughout the study of mathematics and it is important that they are immediately recognizable by children, whatever their magnitude. A variety of possi-ble approaches is suggested here, aimed at developing this competence. TEACHING ODD AND EVEN NUMBERS Very young children are able to construct their own meaning of odd and even numbers. Clarke and Atkinson (1996, p. 55) quote the experience of Derek (age six) when using cubes.
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