OLLSCOIL NA hEIREANN MA NUAD
THE NATIONAL UNIVERSITY OF IRELAND MAYNOOTH
MATHEMATICAL PHYSICS
MP201
Mathematical Methods I
APPENDIX - EVEN AND ODD FUNCTIONS: THEIR
ARITHMETIC AND INTEGRALS
Prof. D. M. Heffernan and Mr. S. Pouryahya
Contents
B Even and Odd Functions: Their Arithmetic and Integrals
B.1 The Definitions . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 Even and Odd Arithmetic . . . . . . . . . . . . . . . . . . .
B.3 Integration of Even and Odd Functions . . . . . . . . . . . .
B.3.1 The integral of an odd function over its domain . . .
B.3.2 The integral of an even function over its domain. . .
B.4 Integrating Products of Even and Odd Functions . . . . . .
B.5 Useful Even and Odd Functions . . . . . . . . . . . . . . . .
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B
Even and Odd Functions: Their Arithmetic and
Integrals
A firm grasp of even and odd functions can be of great use to the reader as the integral
of even and odd functions have properties that can drastically reduce the amount of work
one has to do in order to reach an answer.
B.1
The Definitions
Definition B.1 (Even function).
A function f (x) is said to be even if
f (−x) = f (x)
for every value of x in the domain of definition.
Definition B.2 (Odd function).
A function f (x) is said to be odd if
f (−x) = −f (x)
for every value of x in the domain of definition.
These definitions mean that both the graph of an even function and an odd function have
a certain type of symmetry
The graph of an even function is symmetric about the f (x) axis.
The graph of an odd function is symmetric with respect to the origin.
2
f(x)
f(x)
(x, f(x))
(x, f(x))
(-x, f(-x))
f(x)
-x
x
f(x)
-x
x
x
x
(-x, f(-x))
(a) An example of an even function. Notice
that f (−x) = f (x) for all x.
(b) An example of an odd function. Notice
that f (−x) = −f (x) for all x. The line
through the origin represents the symmetry
present in all odd functions.
Figure 1: Examples of Even and Odd functions.
B.2
Even and Odd Arithmetic
Products of even and odd functions are themselves either even or odd. This is a simple
yet extremely useful result. The arithmetic is as follows
1. The product of two even functions results in an even function
Even × Even = Even
2. The product of an even and an odd functions results in an odd function
Even × Odd = Odd
3. The product of two odd functions results in an even function
Odd × Odd = Even
Quotients of even and odd functions behave in a similar manner taking f1 (x) and f2 (x)
to be odd functions and g1 (x) and g2 (x) to be even functions,
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1. The quotient of two even functions results in an even function
Even
g1 (x)
=
= Even
g2 (x)
Even
2. The quotient of an odd function and an even functions results in an odd function
f1 (x)
Odd
=
= Odd
g1 (x)
Even
Note that g(x) must not be zero on its domain for the quotient to be well defined.
3. The quotient of two odd functions results in an even function
f1 (x)
Odd
=
= Even
f2 (x)
Odd
ADVANCED ASIDE B.1 (Any real function can be considered the sum of an even
and an odd function).
This is in fact an extremely powerful result that we make little use of in this course
(even and odd functions will become much more important at the end of the course
when one examines fourier series), however we include this here as we are examining the
arithmetic of even and odd functions.
Let f (x) be a real functiona for all x ∈ R,
We can write
f (x) = f (x) +
1
[f (−x) − f (−x)]
2
1
= {2f (x) + f (−x) − f (−x)}
2 1
=
f (x) + f (−x) + f (x) − f (−x)
2
1
1
=
f (x) + f (−x) +
f (x) − f (−x)
2
2
= g(x) + h(x).
a
we are not assuming f to be odd or even just a real function.
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. . . advanced aside continued
The idea is to now examine the nature of g(x) and h(x) we have
1
1
f (−x) + f (−{−x}) =
f (−x) + f (x) = g(x)
g(−x) =
2
2
∴ g(x) is Even.
Now examining h(x) we have
1
1
h(−x) =
f (−x) − f (−{−x}) =
f (−x) − f (x) = −h(x)
2
2
∴ h(x) is Odd.
Finally since we have shown that
f (x) = g(x) + h(x)
we have proven that that any real function can be written as a sum of an even and an
odd function.
B.3
Integration of Even and Odd Functions
In this section we will employ the use of the functions f (x) and g(x) now
Let
f (x) be an odd function for −L ≤ x ≤ L
and
g(x) be an even function for −L ≤ x ≤ L.
We are interested in the integral of these functions over their domain ie.
Z+L
f (x) dx and
Z+L
g(x) dx.
−L
−L
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B.3.1
The integral of an odd function over its domain
Important Formula B.1. Given a function f (x) that is odd on the interval −L ≤ x ≤ L
then
ZL
f (x) dx = 0
−L
This result tells us that irrespective of what the function f (x) is, if we know that it is an
odd function over the range that we are integrating over the integral will be equal to zero.
This is due to the way the integral treats areas above and below the axis of integration
(for our functions this is the x-axis). To see this consider the function h(x) that is odd
over its domain, say −π ≤ x ≤ π.
y
y = h(x)
A
-π
π
B
x
Figure 2: Plot of the function h(x) which is odd on the interval −π ≤ x ≤ π. The area
A is the area described by the function above the x axis while the area B is the area
described by the function below the x axis.
Since the h(x) is symmetric about the origin we know that the area the function describes
above the x axis, Area A, and below the x axis, Area B, are the same size. However
the integral makes a distinction between areas above the integration axis and below the
integration axis be means of a factor of −1, so we have
B = −A.
Using this relationship we can integrate the h(x) over −π ≤ x ≤ π
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Zπ
Zπ
Z0
h(x) dx +
h(x) dx =
x=−π
B.3.2
x=0
x=−π
|
h(x) dx = B + A = (−A) + A = 0
{z
Area B
}
|
{z
Area A
}
The integral of an even function over its domain.
Important Formula B.2. Given a function g(x) that is even on the interval −L ≤ x ≤ L
then
ZL
Z0
ZL
g(x) dx = 2
g(x) dx = 2
−L
−L
g(x) dx
0
This can be seen almost immediately by sketching an even function on an interval −L ≤
x ≤ L and looking at the areas on either side of the function axis.
y
y = g(x)
B
A
-L
L
x
Figure 3: Plot of an even function g(x) on the interval −L ≤ x ≤ L.
B.4
Integrating Products of Even and Odd Functions
Given what has now been covered about the integral of even and odd functions and the
arithmetic of even and odd functions, one can turn their attention to the integral of of
products of even and odd functions.
Given f (x) is an odd function and g(x) is an even function, both of which are defined on
the interval −L ≤ x ≤ L then,
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ZL
f (x)g(x) dx = 0.
x=−L
This is because the product of an odd function and an even function results in an odd
function i.e. Odd × Even = Odd, and the integral of an odd function over the interval
of which it is odd is zero.
This method works for any number of products of even and odd functions for example
ZL
g(x)f (x)g(x) dx.
x=−L
Let,
g(x)f (x) = h(x)
Even × Odd = Odd
hence,
g(x)f (x)g(x) = h(x)g(x)
= Odd × Even = Odd
Finally we can say that
ZL
g(x)f (x)g(x) dx = 0.
x=−L
Since the integrand is an odd function over the interval −L ≤ 0 ≤ L.
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B.5
Useful Even and Odd Functions
There are a number of common even and odd functions that are useful to recognise,
f (x) = sin(x) on
−π ≤x≤π
f (x) = sinh(x) on
is Odd.
is Odd.
f(x)
f(x)
−π ≤x≤π
f(x) = sinh(x)
f(x) = sin(x)
-π
π
-π
x
π
Figure 4: The graph of sin(x) over
the interval −π ≤ x ≤ π.
f (x) = cos(x) on
Figure 6: The graph of sinh(x)
over the interval −π ≤ x ≤ π.
−π ≤x≤π
f (x) = cosh(x) on
is Even.
is Even.
f(x)
f(x)
π
−π ≤x≤π
f(x) = cosh(x)
f(x) = cos(x)
-π
x
x
-π
Figure 5: The graph of cos(x) over
the interval −π ≤ x ≤ π.
π
x
Figure 7: The graph of cosh(x)
over the interval −π ≤ x ≤ π.
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f (x) = x on
f (x) = x3
−L≤x≤L
is Odd.
f(x)
f(x)
f(x) = x3
-L
L
-L
x
L
on
x
Figure 10: The graph of x3 over
the interval −L ≤ x ≤ L.
Figure 8: The graph of f (x) = x
over the interval −L ≤ x ≤ L.
f (x) = x4
−L≤x≤L
on
−L≤x≤L
is Even.
is Even.
f(x)
f(x)
f(x) = x2
-L
−L≤x≤L
is Odd.
f(x) = x
f (x) = x2
on
f(x) = x4
L
x
-L
Figure 9: The graph of x2 over the
interval −L ≤ x ≤ L.
L
x
Figure 11: The graph of x4 over
the interval −L ≤ x ≤ L.
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