Periodic Functions

Geometrically, a function is said to be periodic if shifting the function by a fixed amount backward or forward results in the same function. This means that the graph of the function repeats itself identically after such a shift. Mathematically, for a periodic function \( f(x) \), if it is shifted by \( T \) units (backward or forward) along the \( x \)-axis, the resulting graph overlaps perfectly with the original graph, i.e.,

\[ f(x + T) = f(x). \]

where we call \(T\) as the period of the fucntion.

For example, consider the function \( \sin(x) \). From trigonometry, we know that,

\[ \sin(x + 2\pi) = \sin(x), \quad \forall x \in \mathbb{R}. \]

However, this property is not restricted to \( 2\pi \). It is also true for:

\[ \sin(x + 4\pi) = \sin(x), \quad \sin(x + 6\pi) = \sin(x), \quad \sin(x - 2\pi) = \sin(x), \ldots \]

In general, for any integer \( n \in \mathbb{Z} \),

\[ \sin(x + 2n\pi) = \sin(x). \]

Thus, \( 2\pi, 4\pi, 6\pi, -2\pi, \ldots \) are all periods of \( \sin(x) \). However, among these, \( 2\pi \) is the smallest positive value for which this property holds. This smallest positive period is called the fundamental period of the function.

Therefore, for \( \sin(x) \), the fundamental period is \( 2\pi \).

Formal Definition

A function \( f(x) \) is said to be periodic if and only if there exists a positive real number \( T \), called the period of \( f(x) \), such that for every \( x \) in the domain of \( f(x) \):

\( x + T \) also belongs to the domain of \( f(x) \), and
\( f(x + T) = f(x). \)

The smallest positive value of \( T \) that satisfies these conditions is called the fundamental period of \( f(x) \).

The graph represents the periodic function \( f(x) = 3 - ||2x| - 2| \), \(0\leq x \leq 5\) and \(f(x+5) = f(x)\). The function exhibits a repeating pattern every 5 units.

This means that if we shift the graph 5 units forward (\( x \to x+5 \)) or backward (\( x \to x-5 \)), the resulting graph remains identical to the original. The periodicity of the function is \( T = 5 \). The repeating pattern is clearly visible across the \( x \)-axis.

Periodicity for Integer Multiples f(x+nT) = f(x)

If \( f(x+T) = f(x) \) for all \( x \) belonging to the domain of \( f \) and some period \( T \), then \( f(x + nT) = f(x) \) for all \( n \in \mathbb{Z} \).

You can prove it using mathematical induction. We will use this simple reasoning to verify the eracity of this statement.

The identity \( f(x+T) = f(x) \) is given. This property implies that the function \( f(x) \) repeats its values after every interval of length \( T \).

Forward Iteration:

Start with the identity \( f(x+T) = f(x) \).
Replace \( x \) by \( x+T \) in \( f(x+T) = f(x) \), giving:

\[ f((x+T) + T) = f(x+T). \]

This simplifies to:

\[ f(x+2T) = f(x+T). \]

Using \( f(x+T) = f(x) \), we find:

\[ f(x+2T) = f(x). \]

Replace \( x \) again by \( x+T \):

\[ f((x+2T) + T) = f(x+2T). \]

This simplifies to:

\[ f(x+3T) = f(x). \]

Repeating this process for any positive multiple of \( T \), we have:

\[ f(x + nT) = f(x), \quad n \in \mathbb{N}. \]
Backward Iteration:

Replace \( x \) by \( x-T \) in \( f(x+T) = f(x) \):

\[ f((x-T) + T) = f(x-T). \]

This simplifies to:

\[ f(x) = f(x-T). \]

Hence, \( f(x-T) = f(x) \).

Replace \( x \) by \( x-T \) again in \( f(x+T) = f(x) \):

\[ f((x-T) - T) = f(x-T). \]

This simplifies to:

\[ f(x-2T) = f(x-T). \]

Using \( f(x-T) = f(x) \), we get:

\[ f(x-2T) = f(x). \]

Repeating this process for any negative multiple of \( T \), we have:

\[ f(x - nT) = f(x), \quad n \in \mathbb{N}. \]

By repeatedly shifting \( x \) forward or backward by multiples of \( T \), we establish that:

\[ \boxed{f(x + nT) = f(x), \quad \forall n \in \mathbb{Z}} \]

Period Subtraction Property

If \( T_1 \) and \( T_2 \) are both periods of a function \( f(x) \), then \( T_1 - T_2 \) is also a period of \( f(x) \).

Proof:

By definition, if \( T_1 \) and \( T_2 \) are periods of \( f(x) \), we have:

\[ f(x + T_1) = f(x) \quad \text{and} \quad f(x + T_2) = f(x). \]

To show that \( T_1 - T_2 \) is also a period, replace \( x \) with \( x - T_2 \) in the first equation:

\[ f((x - T_2) + T_1) = f(x - T_2). \]

Simplifying the left-hand side:

\[ f(x + T_1 - T_2) = f(x - T_2). \]

Since \( T_2 \) is a period of \( f(x) \), we know:

\[ f(x - T_2) = f(x). \]

Substituting this back, we get:

\[ f(x + T_1 - T_2) = f(x). \]

Thus, \( T_1 - T_2 \) is also a period of \( f(x) \). This reasoning holds for any two periods \( T_1 \) and \( T_2 \), demonstrating the period subtraction property.

Fundamental Periods of Some Elementary Functions

To build a strong understanding of periodic functions, it is essential to know the fundamental periods of commonly used trigonometric functions. The table below summarizes the fundamental periods of these elementary functions:

Six Trigonometric Functions

The six fundamental trigonometric functions—sine, cosine, tangent, cotangent, secant, and cosecant—have well-defined fundamental periods. Below is a concise summary of their periodic behavior:

Function	Fundamental Period
\( \sin x \)	\( 2\pi \)
\( \cos x \)	\( 2\pi \)
\( \tan x \)	\( \pi \)
\( \cot x \)	\( \pi \)
\( \sec x \)	\( 2\pi \)
\( \csc x \)	\( 2\pi \)

Fundamental Periods of the Squares of the Six Trigonometric Functions

When the six trigonometric functions are squared, their periodicity often changes due to the squaring operation, which eliminates sign changes. Here are the fundamental periods of their squares:

\( \sin^2 x \):

The square of the sine function, \( \sin^2 x \), has a fundamental period of \( \pi \). This is because squaring eliminates the negative values of \( \sin x \), causing the function to repeat every half-cycle of \( \sin x \).
\( \cos^2 x \):

Similarly, the square of the cosine function, \( \cos^2 x \), also has a fundamental period of \( \pi \). Like \( \sin^2 x \), squaring the cosine function results in a repeating pattern every half-cycle.
\( \tan^2 x \):

The square of the tangent function, \( \tan^2 x \), retains a fundamental period of \( \pi \). This is because \( \tan x \) itself already repeats every \( \pi \), and squaring does not change this periodicity.
\( \cot^2 x \):

The square of the cotangent function, \( \cot^2 x \), also has a fundamental period of \( \pi \). Like \( \tan^2 x \), squaring does not affect its repeating behavior.
\( \sec^2 x \):

The square of the secant function, \( \sec^2 x \), has a fundamental period of \( \pi \). This is due to the squaring operation and the nature of the vertical asymptotes in the \( \sec x \) function, which repeat every \( \pi \).
\( \csc^2 x \):

The square of the cosecant function, \( \csc^2 x \), also has a fundamental period of \( \pi \). Like \( \sec^2 x \), the periodicity is halved compared to the original \( \csc x \) function.

Fundamental Periods of the Modulus of Trigonometric Functions

When we take the modulus (absolute value) of a trigonometric function, its periodicity often changes because the negative part of the graph is mirrored into the positive part. Below are the fundamental periods of the modulus of the six basic trigonometric functions:

\( |\sin x| \):

The function \( |\sin x| \) is obtained by reflecting the negative half-cycles of \( \sin x \) into the positive half-plane. Since \( \sin x \) repeats every \( 2\pi \), and the modulus halves the cycle, the fundamental period of \( |\sin x| \) is:

\[ T = \pi. \]
\( |\cos x| \):

Similar to \( |\sin x| \), \( |\cos x| \) is obtained by reflecting the negative values of \( \cos x \) into the positive plane. This also halves the periodicity of \( \cos x \), so the fundamental period of \( |\cos x| \) is:

\[ T = \pi. \]
\( |\tan x| \):

The modulus \( |\tan x| \) reflects the negative values of \( \tan x \) into the positive plane. The original fundamental period of \( \tan x \) is \( \pi \), and taking the modulus does not change its periodicity. Thus:

\[ T = \pi. \]
\( |\cot x| \):

The modulus \( |\cot x| \) behaves like \( |\tan x| \), reflecting the negative values into the positive plane. The periodicity remains the same as the original \( \cot x \), so:

\[ T = \pi. \]
\( |\sec x| \):

The function \( |\sec x| \) is already non-negative wherever it is defined because \( \sec x \) does not cross zero. The periodicity of \( \sec x \) remains unchanged:

\[ T = 2\pi. \]
\( |\csc x| \):

Similar to \( |\sec x| \), \( |\csc x| \) is already non-negative wherever it is defined. Its periodicity is the same as the original \( \csc x \):

\[ T = 2\pi. \]

Cube of Trigonometric Functions and Their Periodicity

When we take the cube of trigonometric functions, their periodicity remains the same as the original functions because cubing is an odd operation, which does not eliminate negative values or symmetries. Below are the cubes of the six basic trigonometric functions and their fundamental periods:

\( \sin^3 x \):

Cubing \( \sin x \) preserves its periodicity because it is an odd function and retains the same symmetry as \( \sin x \). The fundamental period remains:

\[ T = 2\pi. \]
\( \cos^3 x \):

Similar to \( \sin^3 x \), \( \cos^3 x \) also retains the symmetry and periodicity of \( \cos x \). The fundamental period is:

\[ T = 2\pi. \]
\( \tan^3 x \):

The cube of \( \tan x \) does not change the periodicity because \( \tan x \) is already an odd function with symmetry about the origin. The fundamental period remains:

\[ T = \pi. \]
\( \cot^3 x \):

Similarly, \( \cot^3 x \) retains the periodicity of \( \cot x \), with symmetry about the origin. The fundamental period is:

\[ T = \pi. \]
\( \sec^3 x \):

Cubing \( \sec x \) does not change its periodicity, as \( \sec x \) retains the same positive and negative patterns. The fundamental period remains:

\[ T = 2\pi. \]
\( \csc^3 x \):

Like \( \sec^3 x \), \( \csc^3 x \) also retains the periodicity of \( \csc x \). The fundamental period is:

\[ T = 2\pi. \]

Cubing does not alter the fundamental periodicity of trigonometric functions, as it preserves their underlying symmetries.

Periods of some non trivial functions

The fundamental period of \(|\sin x|+|\cos x|\) is \(\frac{\pi}{2}\)
1. Plot \( y = |\sin x| \) and \( y = |\cos x| \), reflecting their negative parts above the \( x \)-axis.
2. Add the two graphs pointwise: for every \( x \), calculate \( |\sin x| + |\cos x| \).
3. The resulting graph has a fundamental period of \( \pi \), as both \( |\sin x| \) and \( |\cos x| \) repeat every \( \pi \).
We can also verify that \( f(x) = |\sin x| + |\cos x| \) has a period of \( \frac{\pi}{2} \) as follows:

\[ f(x + \frac{\pi}{2}) = |\sin(x + \frac{\pi}{2})| + |\cos(x + \frac{\pi}{2})|. \]

Using the trigonometric identities:

\[ \sin(x + \frac{\pi}{2}) = \cos x \quad \text{and} \quad \cos(x + \frac{\pi}{2}) = -\sin x, \]

we find:

\[ f(x + \frac{\pi}{2}) = |\cos x| + |-\sin x|. \]

Since \( |a| = |-a| \), this simplifies to:

\[ f(x + \frac{\pi}{2}) = |\cos x| + |\sin x| = f(x). \]

Thus, \( \frac{\pi}{2} \) is a period of \( f(x) \).

To verify that \( \frac{\pi}{2} \) is the fundamental period, we check whether any smaller period exists. If \( \frac{\pi}{2} \) is not the fundamantal period, then \( \frac{\pi}{4} \) must be period as well because \( \frac{\pi}{2} \) is a multiple of \( \frac{\pi}{4} \) \( f(x) \). Then:

\[ f(x + \frac{\pi}{4}) = |\sin(x + \frac{\pi}{4})| + |\cos(x + \frac{\pi}{4})| \]

would need to equal \( |\sin x| + |\cos x| \) for all \( x \). However, using trigonometric identities, this is not true for all \( x \). For example:

\[ \sin(x + \frac{\pi}{4}) = \frac{\sqrt{2}}{2}(\sin x + \cos x), \quad \cos(x + \frac{\pi}{4}) = \frac{\sqrt{2}}{2}(\cos x - \sin x). \]

Taking their absolute values and summing generally does not result in \( |\sin x| + |\cos x| \). Hence, \( \frac{\pi}{4} \) is not a period.

Since \( \frac{\pi}{2} \) is a period and no smaller period exists, we conclude that the fundamental period of \( f(x) = |\sin x| + |\cos x| \) is:

\[ T = \frac{\pi}{2}. \]
Let \( m \in \mathbb{Z} \setminus \{-1, 0, 1\} \). The function \( \sin^{2m}x + \cos^{2m}x \) has the fundamental period \( \frac{\pi}{2} \).
The function \( |\tan x| + |\cot x| \) has the fundamental period \( \frac{\pi}{2} \).
The function \( \tan x \cdot \cot x \) has the fundamental period \( \frac{\pi}{2} \). This can be understood by analyzing its behavior and graph. The expression simplifies to \( 1 \) wherever \( \tan x \) and \( \cot x \) are both defined. However, \( \tan x \) is undefined at \( x = n\pi + \frac{\pi}{2} \) and \( \cot x \) is undefined at \( x = n\pi \), where \( n \in \mathbb{Z} \). Consequently, the graph of \( \tan x \cdot \cot x \) consists of horizontal lines at \( y = 1 \) with holes or missing points at \( x = n\pi \) and \( x = n\pi + \frac{\pi}{2} \). It is evident from the graph that shifting it by \( \frac{\pi}{2} \) results in an identical graph, confirming that the minimum period of the function is \( \frac{\pi}{2} \).
The function \( \tan^{2m}x + \cot^{2m}x \), where \( m \in \mathbb{N} \), has the fundamental period \( \frac{\pi}{2} \). This result holds because both \( \tan^{2m}x \) and \( \cot^{2m}x \) are periodic with period \( \pi \), and their sum exhibits symmetry that reduces the fundamental period to \( \frac{\pi}{2} \).
The functions \( \cos(\cos x) \) and \( \cos(\sin x) \) both have the fundamental period \( \pi \).
\( \cos(\sin x) + \cos(\cos x) \) has the fundamental period \( \frac{\pi}{2} \).
\( \sin(\sin x) \) or \( \sin(\cos x) \) have the fundamental period \( 2\pi \).
\( |\sin(\cos x)| + |\sin(\sin x)| \) has the fundamental period \( \frac{\pi}{2} \).
Fractional Part Function:
- The fractional part function is defined as:
  
  \[ \{x\} = x - \lfloor x \rfloor \]
- It is a periodic function with a fundamental period of \( 1 \).
- The function \( \{x\} + \left\{x - \frac{1}{2}\right\} \) has the fundamental period equal to \( \frac{1}{2} \).
- The function \( \{x\}\{x - \frac{1}{2}\} \) has the fundamental period equal to \( \frac{1}{2} \).

Periodic Functions Can Be Synthesized

A periodic function can be constructed by defining it explicitly in one fundamental interval and then extending it using the property of periodicity:

\[ f(x + T) = f(x), \quad \text{where } T \text{ is the fundamental period}. \]

This recursive approach allows us to define the function compactly over \( \mathbb{R} \).

Example

Consider the function \( f(x) \) defined by:

\[ f(x) = \begin{cases} x & x \in [0, 1), \\ 2 - x & x \in [1, 2), \\ f(x + 2) & x \in \mathbb{R}. \end{cases} \]

This is a periodic function with a fundamental period of 2. The explicit definition over \( [0, 2) \) is:

\[ f(x) = \begin{cases} x & x \in [0, 1), \\ 2 - x & x \in [1, 2). \end{cases} \]

The periodicity implies that the same pattern repeats for all intervals of length 2.

The graph of \( f(x) \) resembles a repeating "V"-shaped pattern. Key features include:

The function linearly increases in \( [0, 1) \), creating an upward slope.
The function linearly decreases in \( [1, 2) \), forming a downward slope.
The pattern repeats periodically across the real line with period 2.

Below is a geometric depiction of \( f(x) \), showing its periodic nature:

The graph oscillates between \( y = 0 \) and \( y = 2 \), with turning points at \( x = n \), \( n \in \mathbb{Z} \).
At \( x = n + \frac{1}{2} \), the function reaches its midpoints \( f(x) = 1 \).

Constant Function

The constant function is periodic, but it does not have a fundamental period. The reason lies in the definition of periodicity and the concept of the fundamental period.

For a constant function \( f(x) = c \) (where \( c \) is a constant), the function satisfies:

\[ f(x + T) = f(x) \quad \text{for all } x \text{ and any real number } T. \]

This means the function repeats itself for any period \( T > 0 \). Since the function is the same at all points, there is no restriction on the value of \( T \). Thus, the constant function is periodic, as it satisfies the condition for periodicity.

However, the fundamental period is defined as the smallest positive value of \( T \) such that \( f(x + T) = f(x) \) for all \( x \). In the case of a constant function, \( T \) can be any positive real number. Since there is no smallest positive real number (due to the density of real numbers), the constant function does not have a fundamental period.

Hence, while the constant function is periodic, it lacks a fundamental period because the condition for having a least positive \( T \) cannot be satisfied.

Algebra of Periodic Functions

Effect of Adding or Multiplying a Non-Zero Constant on Periodicity

If the fundamental period of a function \( f(x) \) is \( T \), then transformations like scaling the function by a non-zero constant \( k \) or shifting it by a constant \( k \) do not affect its periodicity. Specifically, the fundamental periods of \( kf(x) \) and \( f(x) + k \), where \( k \) is a non-zero real number, remain equal to \( T \).

This can be understood as follows:

Scaling by \( k \):

Consider the function \( h(x) = kf(x) \). Since \( f(x + T) = f(x) \) for all \( x \), scaling by \( k \) gives:

\[ h(x + T) = kf(x + T) = kf(x) = h(x). \]

Thus, the periodicity of \( f(x) \) is preserved in \( h(x) \), and the fundamental period remains \( T \).
Shifting by \( k \):

Now consider the function \( g(x) = f(x) + k \). Using the periodicity of \( f(x) \), we have:

\[ g(x + T) = f(x + T) + k = f(x) + k = g(x). \]

Therefore, adding a constant \( k \) does not alter the periodicity, and the fundamental period of \( g(x) \) is still \( T \).

Consider the function \( f(x) = \sin x \), which has a fundamental period of \( 2\pi \). If we scale and shift it to form:

\[ h(x) = 100\sin x - 1000, \]

then:

Scaling by \( 100 \) modifies the amplitude of \( \sin x \), but the period remains unchanged.
Shifting by \( -1000 \) vertically translates the graph, but it does not affect the repeating nature of the function.

Thus, the period of \( h(x) = 100\sin x - 1000 \) is still \( 2\pi \), the same as the period of \( \sin x \).

Effect of Scaling the Input on the Fundamental Period

If the fundamental period of a function \( f(x) \) is \( T \), then scaling the input of the function by a non-zero real number \( k \) results in a new fundamental period equal to \( \frac{T}{|k|} \). In this case, scaling affects the fundamental period directly.

Proof:

Let \( f(x) \) be a periodic function with fundamental period \( T \), so:

\[ f(x + T) = f(x) \quad \text{for all } x. \]

Now, consider \( h(x) = f(kx) \), where \( k \neq 0 \), and let \( T' \) be the fundamental period of \( h(x) \). By the definition of periodicity:

\[ h(x + T') = h(x). \]

Substituting \( h(x) = f(kx) \), we get:

\[ f(k(x + T')) = f(kx). \]

Simplifying this, we obtain:

\[ f(kx + kT') = f(kx). \]

For \( T' \) to be the least positive period of \( h(x) \), the corresponding shift in \( f(x) \) must be the fundamental period \( T \). Hence, we require:

\[ |kT'| = T. \]

Solving for \( T' \), we find:

\[ T' = \frac{T}{|k|}. \]

\(\blacksquare\)

Examples

The fundamental period of \( h(x) = 30\sin(3x - \pi/6) - 12 \) is:

\[ T' = \frac{2\pi}{|3|} = \frac{2\pi}{3}. \]
The function \( f(x) = \left\{\frac{x}{5}\right\} \) (fractional part of \( x/5 \)) has a fundamental period:

\[ T' = \frac{1}{1/5} = 5. \]

Geometrical Interpretation

Since we know that when we scale the input, that is, when we replace \( x \) by \( kx \), the graph undergoes either a compression or a stretching depending on the magnitude of \( k \). If \( |k| > 1 \), the graph is compressed, and the fundamental period becomes smaller. Conversely, if \( 0 < |k| < 1 \), the graph is stretched, and the fundamental period becomes larger.

This is mathematically captured by the relation:

\[ T' = \frac{T}{|k|}, \]

where \( T \) is the fundamental period of the original function, and \( T' \) is the fundamental period of the transformed function. The magnitude of \( k \) inversely affects the fundamental period, compressing the graph when \( |k| > 1 \) and stretching it when \( |k| < 1 \).

Periodicity of the Sum of Two Periodic Functions

When two periodic functions \( f_1(x) \) and \( f_2(x) \) with fundamental periods \( T_1 \) and \( T_2 \) are added, the periodicity of the resulting function \( h(x) = f_1(x) + f_2(x) \) depends on the relationship between \( T_1 \) and \( T_2 \). Below, we rigorously analyze the behavior with examples.

General Rule

The function \( h(x) \) is periodic if and only if the ratio \( \frac{T_1}{T_2} \) is a rational number. In such cases, the period (not the fundamental period) of \( h(x) \) is:

\[ T = \text{LCM}(T_1, T_2), \]

where \( \text{LCM} \) represents the least common multiple of \( T_1 \) and \( T_2 \).

If \( \frac{T_1}{T_2} \) is irrational, \( h(x) \) is aperiodic, meaning it does not repeat after any finite interval. This is because \(\text{LCM}(T_1, T_2)\) does not exist when \( \frac{T_1}{T_2} \) is irrational. Why does it not exist? To understand that one needs to take a look at basic definition of LCM.

Why \(\text{LCM}(T_1, T_2)\) does not exist when \( \frac{T_1}{T_2} \) is irrational?

The LCM of two periods \( T_1 \) and \( T_2 \) is defined as the smallest positive number \( T \) such that:

\[ n_1 T_1 = n_2 T_2, \]

where \( n_1 \) and \( n_2 \) are integers, and \( \gcd(n_1, n_2) = 1 \) (i.e., \( n_1 \) and \( n_2 \) are coprime). Dividing through by \( T_2 \), this implies:

\[ \frac{T_1}{T_2} = \frac{n_2}{n_1}. \]

This relationship shows that \( \frac{T_1}{T_2} \) must be a rational number for \( n_1 \) and \( n_2 \) to exist as integers.

When \( \frac{T_1}{T_2} \) is irrational, it is impossible to find integers \( n_1 \) and \( n_2 \) such that \( n_1 T_1 = n_2 T_2 \). Consequently, no finite \( T \) satisfies the definition of LCM, leading to the conclusion that \( T_1 \) and \( T_2 \) cannot align after any finite number of repetitions. This means the resulting function \( h(x) = f_1(x) + f_2(x) \) does not have a finite repeating interval and is therefore aperiodic.

For example, let \( T_1 = 2\pi \) and \( T_2 = \sqrt{2}\pi \). Here:

\[ \frac{T_1}{T_2} = \frac{2\pi}{\sqrt{2}\pi} = \sqrt{2}, \]

which is irrational. Since \( \text{LCM}(T_1, T_2) \) does not exist, \( h(x) \) is aperiodic and does not repeat after any finite interval. This highlights the impossibility of aligning \( T_1 \) and \( T_2 \) to produce a periodic sum.

Thus, the lack of a least common multiple for irrational ratios \( \frac{T_1}{T_2} \) explains why the sum of two periodic functions becomes aperiodic in such cases.

Example 1: Rational Ratio

Let \( f_1(x) = \sin x \) and \( f_2(x) = \cos(2x) \).

The fundamental period of \( f_1(x) \) is \( T_1 = 2\pi \).
The fundamental period of \( f_2(x) \) is \( T_2 = \pi \).

The ratio \( \frac{T_1}{T_2} = \frac{2\pi}{\pi} = 2 \) is rational. Hence, \( h(x) = \sin x + \cos(2x) \) is periodic, and the period is:

\[ T = \text{LCM}(T_1, T_2) = \text{LCM}(2\pi, \pi) = 2\pi. \]

Example 2: Irrational Ratio

Let \( f_1(x) = \sin x \) and \( f_2(x) = \cos(\sqrt{2}x) \).

The fundamental period of \( f_1(x) \) is \( T_1 = 2\pi \).
The function \( f_2(x) = \cos(\sqrt{2}x) \) has an irrational period \( T_2 = \frac{2\pi}{\sqrt{2}} \).

The ratio \( \frac{T_1}{T_2} = \frac{2\pi}{2\pi / \sqrt{2}} = \sqrt{2} \) is irrational. Therefore, \( h(x) = \sin x + \cos(\sqrt{2}x) \) is aperiodic and does not have a period.

Why is the LCM not necessarilty the fundamental period of the sum?

However, due to symmetry in the behavior of \( |\sin x| \) and \( |\cos x| \), \( h(x) \) repeats after a smaller interval \( T_0 = \frac{\pi}{2} \). Specifically, the symmetry in the absolute values of \( \sin x \) and \( \cos x \) ensures that \( h(x + \frac{\pi}{2}) = h(x) \). Hence, the actual fundamental period of \( h(x) \) is:

\[ T_0 = \frac{\pi}{2}, \]

even though the LCM of \( T_1 \) and \( T_2 \) is \( \pi \). This demonstrates how symmetry in the individual functions can lead to a fundamental period smaller than their LCM.

Now, consider another example where the sum of two periodic functions becomes a constant. Take \( h(x) = \sin^2 x + \cos^2 x \). The fundamental periods of both \( \sin^2 x \) and \( \cos^2 x \) are \( \pi \). However, their sum simplifies to:

\[ h(x) = \sin^2 x + \cos^2 x = 1. \]

Since \( h(x) \) is a constant, it is periodic for every positive \( T \). However, it does not have a fundamental period because there is no smallest positive \( T \) satisfying the periodicity condition.

These examples highlight two important situations where the fundamental period of the sum of two periodic functions may differ from the expected LCM of their individual periods:

Symmetry Leading to a Smaller Period

If the functions \( f_1(x) \) and \( f_2(x) \) exhibit a symmetry such that a smaller period \( T_0 \) satisfies:

\[ f_1(x + T_0) = f_2(x), \quad f_2(x + T_0) = f_1(x), \]

then \( T_0 \) becomes the fundamental period of \( h(x) = f_1(x) + f_2(x) \), even if \( T_0 < \text{LCM}(T_1, T_2) \).
Sum Becoming a Constant

If the sum of two periodic functions simplifies to a constant, then the resulting function is periodic for all positive \( T \). However, it does not have a fundamental period, as no smallest positive \( T \) can exist.

In conclusion, while the LCM of the fundamental periods often provides the fundamental period of the sum of two periodic functions, symmetries or special cases, such as a constant sum, can lead to exceptions. These cases emphasize the importance of analyzing the specific properties of the functions involved.

Determining the Fundamental Period

When evaluating the fundamental period of the sum of two periodic functions \( h(x) = f_1(x) + f_2(x) \) using the LCM formula, it is crucial to verify the result to ensure accuracy. Simply relying on the LCM may lead to incorrect conclusions in cases where symmetries result in a smaller period. Follow these steps to determine the fundamental period:

Start with the LCM Formula

Compute \( T = \text{LCM}(T_1, T_2) \), where \( T_1 \) and \( T_2 \) are the fundamental periods of \( f_1(x) \) and \( f_2(x) \), respectively. This gives an initial candidate for the fundamental period of \( h(x) \).
Verify \( T/2 \)

Check if \( T/2 \) is a period by testing whether:

\[ h(x + T/2) = h(x) \quad \text{for all } x. \]

If \( T/2 \) satisfies this condition, it becomes the new candidate for the fundamental period.
Iterate Further

If \( T/2 \) is found to be a period, test smaller divisions, such as \( T/4 \), \( T/8 \), and so on, until the smallest possible \( T_0 \) is identified. Stop when no further smaller period satisfies the condition.
Conclude the Fundamental Period

If \( T/2 \) (or any smaller division) does not satisfy the periodicity condition, then the original \( T \) computed from the LCM is the fundamental period.

Simplify the Function to Identify the Fundamental Period

Many times, simplifying the given function \( h(x) = f_1(x) + f_2(x) \) can lead to a clearer understanding of its periodicity and result in a more accurate determination of the fundamental period. In fact, a simplified form often reveals symmetries or dependencies that are not immediately apparent in the original expression. Drawing the graph of the function can also assist in this process.

Example: Simplifying \( \sin^4 x + \cos^4 x \)

Consider \( h(x) = \sin^4 x + \cos^4 x \). By using algebraic identities, we can rewrite \( \sin^4 x + \cos^4 x \) as:

\[ \sin^4 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x. \]

Since \( \sin^2 x + \cos^2 x = 1 \), this simplifies further to:

\[ \sin^4 x + \cos^4 x = 1 - 2\sin^2 x \cos^2 x. \]

Using the double-angle identity \( \sin^2 x \cos^2 x = \frac{\sin^2(2x)}{4} \), we can express \( \sin^4 x + \cos^4 x \) as:

\[ \sin^4 x + \cos^4 x = 1 - \frac{\sin^2(2x)}{2}. \]

Now, note that \( \sin^2(2x) \) has a fundamental period of \( \pi \), as \( \sin^2 x \) has a period of \( 2\pi \), and scaling the input by a factor of 2 reduces the period to \( \pi \).

Thus, the fundamental period of \( \sin^4 x + \cos^4 x \) is:

\[ T = \pi. \]

Periodicity of the Product and Quotient of Two Periodic Functions

For two periodic functions \( f_1(x) \) and \( f_2(x) \) with fundamental periods \( T_1 \) and \( T_2 \), the periodicity of their product or quotient depends on the ratio \( \frac{T_1}{T_2} \).

The Product: \( h(x) = f_1(x) \cdot f_2(x) \)

If \( \frac{T_1}{T_2} \) is rational, the product \( h(x) \) is periodic, with the period given by:

\[ T = \text{LCM}(T_1, T_2). \]

This happens because the cycles of \( f_1(x) \) and \( f_2(x) \) align after a common multiple of their periods. For example, \( h(x) = \sin x \cdot \cos x \). Here, \( \sin x \) and \( \cos x \) both have a fundamental period of \( 2\pi \), so the LCM is \( 2\pi \), making \( h(x) \) periodic with \( T = 2\pi \).

If \( \frac{T_1}{T_2} \) is irrational, the product \( h(x) \) is aperiodic. This occurs because the cycles of \( f_1(x) \) and \( f_2(x) \) never align after any finite interval, making \( h(x) \) non-repeating.

The Quotient: \( h(x) = \frac{f_1(x)}{f_2(x)} \)

For quotients, the periodicity follows the same logic as the product. If \( \frac{T_1}{T_2} \) is rational, \( h(x) \) is periodic with:

\[ T = \text{LCM}(T_1, T_2). \]

If \( \frac{T_1}{T_2} \) is irrational, the quotient \( h(x) \) is aperiodic for the same reasons as the product.

While the least common multiple (LCM) of the fundamental periods \( T_1 \) and \( T_2 \) of two periodic functions \( f_1(x) \) and \( f_2(x) \) often provides a candidate for the fundamental period of their product or quotient, it is important to note that this is not always the case. The actual fundamental period may depend on the symmetry of the functions and their interactions.

Consider the product \( h(x) = \{x\} \cdot \{x - \frac{1}{2}\} \), where \( \{x\} \) denotes the fractional part of \( x \):

The fundamental period of \( \{x\} \) is \( T_1 = 1 \), as \( \{x\} = x - \lfloor x \rfloor \) repeats every integer.
Similarly, the fundamental period of \( \{x - \frac{1}{2}\} \) is \( T_2 = 1 \).
The LCM of \( T_1 \) and \( T_2 \) is \( T = 1 \).

However, due to the symmetry in the product \( h(x) = \{x\} \cdot \{x - \frac{1}{2}\} \), the actual fundamental period is \( T_0 = \frac{1}{2} \). This smaller period arises because the product repeats every half-unit interval.

Another example is the product \( h(x) = \sin x \cdot \cos x \):

The fundamental period of \( \sin x \) is \( T_1 = 2\pi \).
The fundamental period of \( \cos x \) is \( T_2 = 2\pi \).
The LCM of \( T_1 \) and \( T_2 \) is \( T = 2\pi \).

However, the product simplifies using the double-angle identity:

\[ \sin x \cdot \cos x = \frac{\sin(2x)}{2}. \]

While the LCM provides a useful starting point for determining the fundamental period, simplifying the expression, analyzing the graph, or testing smaller intervals like \( T/2, T/4, \dots \) is essential for accuracy.

Generalizing Periodicity for Expressions Involving Sum, Product, and Quotient

When an expression is composed of multiple terms combined using sums, products, and quotients of periodic functions, the determination of the fundamental period requires careful analysis. The general strategy involves calculating the fundamental period of each individual term and then considering their interactions.

General Rule

If an expression \( h(x) \) is composed of multiple terms \( f_1(x), f_2(x), \dots, f_n(x) \) through sums, products, or quotients, the fundamental period \( T \) of \( h(x) \) is typically a candidate based on the least common multiple:

\[ T = \text{LCM}(T_1, T_2, \dots, T_n), \]

where \( T_1, T_2, \dots, T_n \) are the fundamental periods of the individual terms. However, \( T \) may not always be the fundamental period, as symmetries or cancellations in the expression can result in a smaller period.

Example

Prove that the fundamental Period of \( f(x) = \{x\} \{x - \frac{1}{2}\} \) is \(\frac{1}{2}\)

The function \( f(x) = \left\{x\right\} \left\{x - \frac{1}{2}\right\} \), where \( \left\{x\right\} \) denotes the fractional part of \( x \), requires determining its fundamental period.

Step 1: Periods of Individual Terms

The fractional part \( \left\{x\right\} \) has a fundamental period of \( T_1 = 1 \), and \( \left\{x - \frac{1}{2}\right\} \) also has a fundamental period of \( T_2 = 1 \). Using the LCM rule, the candidate fundamental period is:

\[ T = \text{LCM}(T_1, T_2) = \text{LCM}(1, 1) = 1. \]

Step 2: Testing \( T/2 = \frac{1}{2} \) as a Period

To verify whether \( T/2 = \frac{1}{2} \) is a period, calculate \( f(x + \frac{1}{2}) \):

\[ f(x + \frac{1}{2}) = \left\{x + \frac{1}{2}\right\} \left\{x + \frac{1}{2} - \frac{1}{2}\right\}. \]

Simplify each term:

\[ f(x + \frac{1}{2}) = \left\{x + \frac{1}{2}\right\} \left\{x\right\}. \]

Using the property of fractional parts \( \left\{x + n\right\} = \left\{x\right\} \) for any integer \( n \), rewrite \( \left\{x + \frac{1}{2}\right\} \) as:

\[ \left\{x + \frac{1}{2}\right\} = \left\{x - \frac{1}{2} + 1\right\} = \left\{x - \frac{1}{2}\right\}. \]

Substituting back, we get:

\[ f(x + \frac{1}{2}) = \left\{x - \frac{1}{2}\right\} \left\{x\right\} = f(x). \]

Thus, \( T/2 = \frac{1}{2} \) is a period of \( f(x) \).

Step 3: Testing \( T/4 = \frac{1}{4} \) as a Period

Next, check if \( T/4 = \frac{1}{4} \) is a period:

\[ f(x + \frac{1}{4}) = \left\{x + \frac{1}{4}\right\} \left\{x + \frac{1}{4} - \frac{1}{2}\right\}. \]

Simplify:

\[ f(x + \frac{1}{4}) = \left\{x + \frac{1}{4}\right\} \left\{x - \frac{1}{4}\right\}. \]

\[ f(x + \frac{1}{4}) \neq f(x). \]

Thus, \( T/4 = \frac{1}{4} \) is not a period.

The smallest period that satisfies \( f(x + T_0) = f(x) \) for all \( x \) is \( T_0 = \frac{1}{2} \). Hence, the fundamental period of \( f(x) = \left\{x\right\} \left\{x - \frac{1}{2}\right\} \) is:

\[ T_0 = \frac{1}{2}. \]

Rules Regarding Composition of Periodic Functions

Consider a periodic function \( g(x) \) with fundamental period \( T \). If \( g(x) \) is composed with another function \( f \), resulting in \( f(g(x)) \), then \( f(g(x)) \) is periodic with period \( T \). However, we emphasize that this period \( T \) is not necessarily the fundamental period of \( f(g(x)) \); it must be verified separately. The function \( f \) itself can also influence the periodicity. Remember the effect of modulus on the periodicity of sine.

Examples

Exponential Function:
- \( g(x) = \sin x \), \( f(y) = e^y \).
- \( e^{\sin x} \) is periodic with fundamental period \( 2\pi \), since \(\sin x\) is periodic with period \( 2\pi \) and \( e^y \) preserves this periodicity.
Fractional Part:
- \( g(x) = \{x\} \), where \( \{x\} \) denotes the fractional part function with fundamental period \( 1 \).
- \( \sin \{x\} \) is periodic with fundamental period \( 1 \), since \(\{x\}\) repeats every \( 1 \).
Double Cosine:
- \( g(x) = \cos x \), \( f(y) = \cos y \).
- \( \cos(\cos x) \) is periodic with fundamental period \( \pi \), not \( 2\pi \), because outer \(\cos\) influeced the periodicity.
Square Root of Sine:
- \( g(x) = \sin x \), \( f(y) = \sqrt{y} \) (valid for \( y \geq 0 \)).
- \( \sqrt{\sin x} \) is periodic with fundamental period \( 2\pi \), as the domain restriction of \( \sin x \geq 0 \) does not alter the periodicity of \( \sin x \).

Composition of a Periodic with a aperiodic Function

When \( f \) is periodic with period \( T \), and \( g \) is aperiodic (not periodic), the composition \( f(g(x)) \) is typically aperiodic. However, there are specific exceptions where \( f(g(x)) \) can still be periodic, depending on the structure of \( g(x) \). Understanding and identifying these exceptions is crucial.

When \( f \) is periodic and \( g(x) \) is a linear function of the form \( ax+b \) (which is aperiodic), then \( f(g(x)) \) is periodic with period \( \frac{T}{|a|} \), where \( T \) is the period of \( f(x) \).

Another interesting example where \( f \) is periodic, \( g(x) \) is aperiodic, yet \( f(g(x)) \) is periodic is \( \sin(\pi[x]) \), where \( [x] \) is the greatest integer function. Here, \( \sin(\pi[x]) \) has period \( 2 \).

To verify:

\[ \sin(\pi[x+2]) = \sin(\pi([x]+2)) = \sin(\pi[x] + 2\pi). \]

Using the property that \( \sin(\theta+2\pi) = \sin(\theta) \), we get:

\[ \sin(\pi[x+2]) = \sin(\pi[x]). \]

Hence, \( \sin(\pi[x]) \) is periodic with period \( 2 \).