The fundamental theorem of simple harmonic functions

By Safwan R. Arekat


The defining relation for a one-dimensional real-valued simple harmonic function designated as (pronounced “scene” and symbolized ) is introduced as . It is theorized that this definition of the derivative as the function itself translated on the real line by a certain positive number is a sufficient condition for proving that also satisfies the one-dimensional oscillator equation . A proof of this hypothesis is provided by establishing the continuity, differential, and boundedness properties of and all of its higher-order derivatives and antiderivatives, while relying critically on Roe’s characterization of the sine function. The function enables the adoption of independent non-circular definitions of the trigonometric functions sine and cosine. The properties of are investigated, and trigonometric symmetries and identities are derived directly from the defining relation and its corollaries. A formulation for the unique solution of the function is proposed. Several useful operational theorems are stated and proved. The function is applied to solve problems in trigonometry and physics.

1. Introduction

The importance of oscillations in science and technology cannot be overestimated. Indeed, the subject is so familiar to mathematicians, physicists, and engineers that it needs little introduction. While their origin extends back to antiquity in describing triangles and geometry, the sine and cosine functions have evolved in concept, definition, and usage. They constitute the simplest mathematical description of spatial and temporal periodic oscillations, namely the simple harmonic function. It was Leonhard Euler who first recognized that the simple harmonic motion of a forced mechanical oscillator could be described by transcendental functions of time which he defined analytically to be the sine and the cosine functions Reference 1. The properties and the identities of these functions were not necessarily determined from line segments, circles, or triangles any longer, but from the defining differential equations Reference 2. This ushered in the mathematical techniques of analytical trigonometry. The consistency between the analytically obtained and the geometrically obtained properties was convincing of the validity of the new functions, and very satisfying as well. Later, Euler advanced the functions further by establishing their close and useful relationship with the complex exponential function. Powerful techniques based on analytical trigonometry, the Fourier theorems in particular, found immense applicability in mathematics and scientific fields. The sine and the cosine functions have become deeply rooted central paradigms in mathematics.

The history of mathematics is characterized by an incremental evolution in methods and concepts. Even in recent times, the characterization of the trigonometric functions continues to advance toward completeness. A few decades ago, Roe Reference 3 had arguably provided the most important contribution to advancing our understanding of trigonometric functions in a very long time. Roe’s characterization can be summarized as follows. Any function on the real line with the property that all its derivatives and antiderivatives are uniformly bounded must be a solution to the one-dimensional oscillator equation

which is necessarily satisfied by simple harmonic oscillations with period . This striking argument for characterization by boundedness was hypothesized and proved by Roe and expanded by other studies. Most notably, Strichartz Reference 4 generalized it for higher dimensions and Howard Reference 5 made it encompass complex functions. We quote Roe’s theory below because it is critically important for our own theory. In his paper Reference 3, Roe proves the following.

Theorem 1.1.

Let be a two-way infinite sequence of real-valued functions defined on the real line  . Assume that and that there is a constant M such that for all n and x. Then for some real constants and .

The present paper continues the effort toward a completeness of our understanding of trigonometric functions. Herein, the concepts of sine and cosine functions are re-evaluated through the fine-tuning of their defining relations. We show that one of the properties of the sine and cosine functions, namely their translational symmetry, can be promoted to a fundamental defining relation. In order to avoid confusion with the longstanding notions of sine and cosine, we define a new primary function (pronounced “scene” and symbolized ). The sine and cosine functions can then be re-defined as translations of this function.

The main aim in the present paper is to prove that the real-valued function with the property

satisfies the conditions in Roe’s theorem to qualify as a solution to the oscillator equation Equation 1.1. We use a slightly different formulation from that used by Roe, but eventually we recast it into Roe’s notation for comparison. We investigate the numerical and symmetry properties of the simple harmonic function, relating it to the trigonometric functions sine and cosine, and we derive trigonometric identities and properties relying only on the fundamental relation and its development, without resorting to geometry, power series expansions, or complex exponential functions. We then propose a formulation for that is suitable for practical calculations, apply it to two selected problems, and discuss the advantages of using this new function.

2. The theorem

Theorem 2.1 (The fundamental theorem of simple harmonic functions).

Let the real-valued function be defined on the real line , and let it satisfy the relation for and . Then


To prove that is compatible with Roe’s theory, we must investigate the continuity, differentiability, antidifferentiability, and boundedness of and its differentially related functions.

Lemma 2.2.

The function is infinitely differentiable; .


The fundamental relation is an explicit statement that is differentiable to first order, with the derivative equal to the function itself translated by a positive real number . Because is differentiable, it is straightforward to conclude by the chain rule that is differentiable as well. Therefore, the derivative is further differentiable to give the second-order derivative . The existence of all higher-order derivatives can be proved by an iteration of this argument. In general,

where is any positive integer. The order-m derivative of becomes an m-fold translation by of on the real line. For a positive value of , differentiation corresponds to shifting to the left. We therefore conclude that all higher-order derivatives exist and are continuous.

Lemma 2.3.

The function is infinitely integrable, with continuous antiderivatives


The function is differentiable, therefore it is continuous. This is a sufficient condition for the existence of its antiderivative, which is simply the indefinite integral of , . We can transform the fundamental relation by subtracting from its argument such that , giving

where is a constant of integration. Because it is a translation of the function itself by , the antiderivative of is continuous, allowing for successive integration. The second-order antiderivative is

The existence of all higher-order antiderivatives can be proved through an iteration of the functional-translation integration procedure, while recognizing the continuity of the integration polynomials. In general, we have that

Dropping the integration polynomial by setting all constants to zero is necessary for proving the boundedness of a subset of antiderivatives, this being because polynomials in general are not bounded on . However, this elimination does not contradict the main proposition of the existence of continuous antiderivatives of all orders. In the remainder of this paper, we refer to this unique subset of antiderivatives with eliminated polynomials as the antiderivatives of . Therefore, the order-m antiderivative is an m-fold translation by of on the real line. For a positive value of , antidifferentiation corresponds to shifting to the right. We therefore conclude that all higher-order antiderivatives exist and are continuous.

Lemma 2.4.

The set of real functions containing and all of its higher-order derivatives and antiderivatives is uniformly bounded on .


Let the real line  be partitioned serially into adjacent closed intervals , where is an integer ranging from to , with considered to be positive. The intervals overlap on their boundaries and have a uniform width . Therefore, all points on the real line are contained in at least one interval in the set. We begin by studying the properties of the functions in the set in the centrally positioned closed interval . Lemmas 2.2 and 2.3 prove the existence of derivatives and antiderivatives for of any order and that are continuous on  and hence continuous on . The boundedness theorem demands that a continuous function defined on a closed interval be bounded on that interval. It follows that and all of its higher derivatives and antiderivatives – namely the elements of – are bounded on the chosen interval . However, this is only the first step because we must also prove the boundedness of the functions in in all other intervals on the real line. The boundedness theorem could be used again to prove the boundedness of on intervals in the neighborhood of . However, this argument breaks down as , without having to consider further the details of the functions in . Let be any point in the interval , being related to point through a translation by as

Now, the order-n derivative of , belonging to which is given by equation Equation 2.1 is evaluated at as

and the order-n antiderivative of , belonging to which is given by equation Equation 2.2 is evaluated at as

The crucial argument for boundedness is that the function in any interval outside the central interval can always be equated to a particular function in at a point inside , where the boundedness of all the functions in has already been proved. This is enabled via the translational definitions of the derivatives and antiderivatives. In particular, for , the value of in any interval to the right of can be equated to the order-n derivative inside . Because is bounded on , we can conclude that must also be bounded on . Similarly, for , the value of in any interval to the left of can be equated to the order-n antiderivative evaluated inside . We can similarly conclude that because is bounded on , then must also be bounded on . Because ranges from to , the boundedness of on any interval on the real line can be proved by iterating the values of n. Therefore, the function is bounded on the entire real line. We define , a positive real number, as the absolute bound of the function , namely

The proof is concluded by restating that the functions contained in are the function and all its derivatives and antiderivatives, which are no more than translations of itself on the real line. Translational symmetry preserves functional shape, thus maintaining the absolute bound M for all the functions in . This can be written as and and , and we can finally conclude that the set of functions is uniformly bounded on .

For comparison with Roe’s theory, we reformulate our conclusion in the notation of Roe as follows. By combining the results of equations Equation 2.3 and Equation 2.4, we have

with n being any positive or negative integer or zero. We can write the elements of as a two-way infinite sequence of real-valued functions defined on the real line  with and satisfying

The notation specifies that is the antiderivative of and that the derivatives and antiderivatives of exist to any order. Specifically, the antiderivatives of are given by the negative values of n, and its derivatives are given by the positive values of n. The sequence is uniformly bounded on and there exists a constant M such that for all and ,

We can finally conclude that the function has all the properties of Roe’s theorem and is therefore a solution of equation Equation 1.1

To complete the proof of our theorem, we must find the numerical value of . By differentiating the above equation twice more, we obtain the fourth-order derivative

meaning that is a periodic function with period . The period of the solution to the second-order equation Equation 1.1 is , thereby giving . However, other values of p are possible because of the periodicity of . This concludes the proof of our theorem.

3. Properties of the function

In the following discussion, we will interchangeably use the notation and the prime notation to indicate differtiation. However, the prime notation is used to represent differentiation with respect to the argument of a function. For example, and , where is a function of the variables , , etc. Also, for brevity we continue to use the symbol instead of .

3.1. Simple harmonic first-order differential equations

The simple harmonic function is defined to satisfy

Together with the second-order oscillator equation,

must also satisfy the first-order equations

3.2. Relation to trigonometric functions

is a general solution to the second-order oscillator equation and is similar to , which is a linear combination of and . Our hypothesis allows to be defined by a single statement, whereas two statements are needed to define the trigonometric functions sine and cosine. The function is the root function for both, and there seems to be an unnecessary redundancy in the need for two defining statements. Traditionally, sine and cosine are defined as satisfying the coupled first-order equations

with boundary conditions (BC’s) 0 and .

The equations are decoupled in second order such that and each satisfy the second-order differential equation. Novel definitions for and are introduced by comparing the first-order equations Equation 3.1 and Equation 3.3 to the standard statements for and above, respectively. By inspection, the trigonometric functions can be restated as

with BC’s and . What is novel here is that both and can now be defined independently of each other, in a direct non-circular fashion, in terms of a single function whose argument is translated by between them. Furthermore, according to this development, one could eliminate and altogether and replace them with only functions. However, it is one thing to demonstrate unified theoretical definitions of the trigonometric functions, but advocating a reformulation is a different matter altogether. The trigonometric functions have become enormously rooted mathematical paradigms, and only overwhelming analytical, algorithmic, or numerical advantage might make such a reformulation conceivable. In this paper, the function itself is the object of investigation, so it is given precedence over the trigonometric functions.

Having the foresight of conventional trigonometric functions and their properties will guide the derivation of the properties and symmetries of . However, this will be pursued from the defining differential equations and the properties that are established along the way, without the direct use of trigonometric properties, whether geometric, power series or complex exponential relations.

3.3. Boundary conditions

The existence and uniqueness theorem guarantees unique solutions to linear second-order ordinary differential equations. These are determined by specifying the value of the function and its first derivative at a given point. Since the first derivative of is the function itself shifted by , finding the unique solution for depends on specifying the values of at any two points on the real line that are separated by . The expression for the function written as does not possess the free parameters necessary to specify unique solutions. For now, it is instructive from a theoretical standpoint to refrain from the numerical assignment of the BC’s. Greater generality in deriving the properties and symmetries of may be achieved by recognizing the need for some definite (but unspecified) BC’s. A suitable choice for the BC points could be and , in which case we give the BC’s symbolically as and . This contrasts the conventional approach of needing numerical assignments for the BC’s for the and in order to deduce their properties.

3.4. Periodicity

As derived above, is a periodic function with period , namely

3.5. Translational antisymmetry

is antisymmetric for a translation of its argument by . By differentiating twice, we find that

3.6. Equidistant extrema and zero crossings

Let the points , , and be a sequence of three points for an extremum, a zero crossing, and an extremum of , respectively. By repeated differentiation with respect to its argument, we find that

In other words, the argument increases by a uniform value of along the sequence.

3.7. Linear independence

The functions and constitute a pair of linearly independent functions on the real line. To show this, assume that they are linearly related by a constant such that . Differentiating this gives , giving , thereby contradicting the assumption that is real.

3.8. The sum of and is constant

Taking the first derivative, of the sum of squares, we obtain

3.9. Arbitrary solutions to the simple harmonic differential equations

Any arbitrary function that satisfies the second-order equation is a linear combination of and , written as

where and are any real numbers.

3.10. The central theorem of trigonometry (the sum angle relation)

The relation is the generalization of the sum of angles for the trigonometric functions, being the central theorem of trigonometry Reference 2. Here and are two independent real variables. Using linear independence, and assuming the existence of differentiable functions and , we write the expression as

Taking the first and second partial derivatives of this expression with respect to and gives