EULER ‘S THEOREM A function homogeneous of some degree has a property sometimes used in economic theory that was first discovered by Leonhard Euler (1707-1783).

Proposition (Euler's theorem) The differentiable function  f  of n variables is homogeneous of degree k if and only if ∑i=1

nxi f i'(x1... xn) = k f (x1... xn) for all (x1... xn). (*)

Condition (*) may be written more compactly, using the notation ∇ f  for the gradient vector of  f  and letting x = (x1, ..., xn), as

x·∇ f (x) = k f (x) for all x.

Proof I first show that if  f  is homogeneous of degree k then (*) holds. If  f  is homogeneous of degree k then

 f (tx1, ..., txn) = tk f (x1, ..., xn) for all (x1, ..., xn) and all t > 0.Differentiate each side of this equation with respect to t, to give x1 f '1(tx1, ..., txn) + x2 f '2(tx1, ..., txn) + ... + xn f 'n(tx1, ..., txn) = ktk−1 f (x1, ..., xn).Now set t = 1, to obtain (*).

I now show that if (*) holds then  f  is homogeneous of degree k. Suppose that (*) holds. Fix (x1, ..., xn) and

define the function g of a single variable by

g(t) = t−k f (tx1, ..., txn) −  f (x1, ..., xn).We have g'(t) = −kt−k−1 f (tx1, ..., txn) + t−k∑i=1

n xi f 'i(tx1, ..., txn).By (*), we have ∑i=1

ntxi f i'(tx1, ..., txn) = k f (tx1, ..., txn),so that g'(t) = 0 for all t. Thus g(t) is a constant. But g(1) = 0, so g(t) = 0 for all t, and hence  f (tx1, ..., txn) = tk f (x1, ..., xn) for all t > 0, so that  f  is homogeneous of degree k.

Example Let  f (x1... xn) be a firm's production function; suppose it is homogeneous of degree 1 (i.e. has "constant returns to scale"). Euler's theorem shows that if the price (in terms of units of output) of each input i is its "marginal product"  f 'i(x1, ..., xn), then the total cost, namely ∑i=1

nxi f i'(x1, ..., xn)is equal to the total output, namely  f (x1, ..., xn).

If there exists an, such that Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x,

where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine

and sine, with the argument x given in radians rather than in degrees. The formula is still valid if x is a complex number, and so some authors refer to the more general complex version as Euler's formula.[1]

Richard Feynman called Euler's formula "our jewel" and "the most remarkable formula in mathematics".[2]

Homogeneous functions

Definition

Multivariate functions that are "homogeneous" of some degree are often used in economic theory. A function is homogeneous of degree k if, when each of its arguments is multiplied by any number t > 0, the value of the function is multiplied by tk. For example, a function is homogeneous of degree 1 if, when all its arguments are multiplied by any number t > 0, the value of the function is multiplied by the

same number t.

Here is a precise definition. Because the definition involves the relation between the value of the function at (x1, ..., xn) and

it value at points of the form (tx1, ..., txn) where t is any positive number, it is restricted to functions for which (tx1, ..., txn) is in the domain whenever t > 0 and (x1, ..., xn) is in the domain. (Some domains that have this property are: the set of all real numbers, the set of nonnegative real numbers, the set of positive real numbers, the set of all n-tuples (x1, ..., xn) of real numbers, the set of n-tuples of nonnegative real numbers, and the set of n-tuples of positive real numbers.)

Definition A function  f  of n variables for which (tx1, ..., txn) is in the domain whenever t > 0 and (x1, ..., xn) is in the domain is homogeneous of degree k if  f (tx1... txn) = tk f (x1, ..., xn) for all (x1, ..., xn) in the domain of  f  and all t > 0.

Example For the function  f (x1, x2) = Ax1

ax2b with domain {(x1, x2):

x1 ≥ 0 and x2 ≥ 0} we have  f (tx1, tx2) = A(tx1)a(tx2)b = Ata+bx1

ax2b = ta+b f (x1, x2),

so that  f  is homogeneous of degree a + b.

Example Let  f (x1, x2) = x1 + x2

2, with domain {(x1, x2): x1 ≥ 0 and x2 ≥ 0}. Then  f (tx1, tx2) = tx1 + t2x2

2.It doesn't seem to be possible to write this expression in the form tk(x1 + x2

2) for any value of k. But how do we prove that there is no such value of k? Suppose that there were such a value. That is, suppose that for some k we have

tx1 + t2x22 = tk(x1 + x2

2) for all (x1, x2) ≥ (0, 0) and all t > 0.Then in particular, taking t = 2, we have 2x1 + 4x2 = 2k(x1 + x2

2) for all (x1, x2).Taking (x1, x2) = (1, 0) and (x1, x2) = (0, 1) we thus have 2 = 2k and 4 = 2k,which is not possible. Thus  f is not homogeneous of any degree.

In economic theory we often assume that a firm's production function is homogeneous of degree 1 (if all inputs are multiplied by t then output is multiplied by t). A production function with this property is said to have "constant returns to scale".

Suppose that a consumer's demand for goods, as a function of prices and her income, arises from her choosing, among all the bundles she can afford, the one that is best according to her preferences. Then we can show that this demand function is homogeneous of degree zero: if all prices and the consumer's income are multiplied by any number t > 0 then her demands for goods stay the same.

Partial derivatives of homogeneous functions:The following result is sometimes useful.

Proposition Let  f  be a differentiable function of n variables that is homogeneous of degree k. Then each of its partial

derivatives  f 'i (for i = 1, ..., n) is homogeneous of degree k − 1.

Proof The homogeneity of  f  means that  f (tx1, ..., txn) = tk f (x1, ..., xn) for all (x1, ..., xn) and all t > 0.Now differentiate both sides of this equation with respect to xi, to get t f 'i(tx1, ..., txn) = tk f 'i(x1, ..., xn),and then divide both sides by t to get  f 'i(tx1, ..., txn) = tk−1 f 'i(x1, ..., xn),so that  f 'i is homogeneous of degree k − 1.

Application: level curves of homogeneous functions

This result can be used to demonstrate a nice result about the slopes of the level curves of a homogeneous function. As we have seen, the slope of the level curve of the function F  through the point (x0, y0) at this point is

−

F 1'(x0, y0)

F 2'(x0, y0).

Now suppose that F  is homogeneous of degree k, and consider the level curve through (cx0, cy0) for some number c > 0. At (cx0, cy0), the slope of this curve is

−

F 1'(cx0, cy0)

F 2'(cx0, cy0).

By the previous result, F '1 and F '2 are homogeneous of degree k−1, so this slope is equal to

−

ck−1F 1'(x0, y0)

ck−1F 2'(x0, y0)= −

F 1'(x0, y0)

F 2'(x0, y0).

That is, the slope of the level curve through (cx0, cy0) at the point (cx0, cy0) is exactly the same as the slope of the level curve through (x0, y0) at the point (x0, y0), as illustrated in the following figure.

In this figure, the red lines are two level curves, and the two green lines, the tangents to the curves at (x0, y0) and at (cx0, xy0), are parallel.

We may summarize this result as follows.

Let F  be a differentiable function of two variables that is homogeneous of some degree. Then along any given ray from the origin, the slopes of the level curves of F  are the same.

Contents 1 History 2 Applications in complex number theory 3 Relationship to trigonometry 4 Other applications 5 Definitions of complex exponentiation

o 5.1 Taylor series definition o 5.2 Analytic continuation definition o 5.3 Limit definition o 5.4 Differential equation definition o 5.5 Multiplicative property definition

6 Proofs o 6.1 Using Taylor series

o 6.2 Using calculus o 6.3 Using ordinary differential equations

7 See also 8 References 9 External links

HistoryEuler's formula was proven for the first time by Roger Cotes in 1714 in the form

(where "ln" means natural logarithm, i.e. log with base e).

It was Euler who published the equation in its current form in 1748, basing his proof on the infinite series of both sides being equal. Neither of these men saw the geometrical interpretation of the formula: the view of complex numbers as points in the complex plane arose only some 50 years later (see Caspar Wessel). Euler considered it natural to introduce students to complex numbers much earlier than we do today.

In his elementary algebra text book, Elements of Algebra, he introduces these numbers almost at once and then uses them in a natural way throughout.

Applications in complex number theory

This formula can be interpreted as saying that the function eix traces out the unit circle in the complex number plane as x ranges through the real numbers. Here, x is the angle that a line connecting the origin with a point on the unit circle makes with the positive real axis, measured counter clockwise and in radians.

The original proof is based on the Taylor series expansions of the exponential function ez (where z is a complex number) and of sin x and cos x for real numbers x (see below). In fact, the same proof shows that Euler's formula is even valid for all complex numbers z.

A point in the complex plane can be represented by a complex number written in cartesian coordinates. Euler's formula provides a means of conversion between cartesian coordinates and polar coordinates. The polar form reduces the number of terms from two to one, which simplifies the mathematics when used in multiplication or powers of complex numbers. Any complex number z = x + iy can be written as

where

the real part the imaginary part the magnitude of z atan2(y, x)

φ is the argument of z—i.e., the angle between the x axis and the vector z measured counterclockwise and in radians—which is defined up to addition of 2π.

Now, taking this derived formula, we can use Euler's formula to define the logarithm of a complex number. To do this, we also use the definition of the logarithm (as the inverse operator of exponentiation) that

and that

Therefore, one can both valid for any complex numbers a and b.

write:

for any . Taking the logarithm of both sides shows that:

and in fact this can be used as the definition for the complex logarithm. The logarithm of a complex number is thus a multi-valued function, because φ is multi-valued.

Finally, the other exponential law

which can be seen to hold for all integers k, together with Euler's formula, implies several trigonometric identities as well as de Moivre's formula.

Relationship to trigonometry

Euler's formula provides a powerful connection between analysis and trigonometry, and provides an interpretation of the sine and cosine functions as weighted sums of the exponential function:

The two equations above can be derived by adding or subtracting Euler's formulas:

and solving for either cosine or sine.

These formulas can even serve as the definition of the trigonometric functions for complex arguments x. For example, letting x = iy, we have:

Complex exponentials can simplify trigonometry, because they are easier to manipulate than their sinusoidal components. One technique is simply to convert sinusoids into equivalent expressions in terms of exponentials. After the manipulations, the simplified result is still real-valued. For example:

Another technique is to represent the sinusoids in terms of the real part of a more complex expression, and perform the manipulations on the complex expression. For example:

This formula is used for recursive generation of cos(nx) for integer values of n and arbitrary x (in radians).

3-D vector:  [x, cos(x), sin(x)]  traces a helix along x axis

Other applicationsIn differential equations, the function eix is often used to simplify derivations, even if the final answer is a real function involving sine and cosine. Euler's identity is an easy consequence of Euler's formula.

In electrical engineering and other fields, signals that vary periodically over time are often described as a combination of sine and cosine functions (see Fourier analysis), and these are more conveniently expressed as the real part of exponential functions with imaginary exponents, using Euler's formula. Also, phasor analysis of circuits can include Euler's formula to represent the impedance of a capacitor or an inductor.

Definitions of complex exponentiationMain articles: Exponentiation and Exponential function

In general, raising e to a positive integer exponent has a simple interpretation in terms of repeated multiplication of e. Raising e to zero or a negative integer exponent can be understood as repeated division. A rational number exponent can be defined by radicals of e, and an irrational number exponent can be defined by finding rational-number exponents that are arbitrarily close to the irrational-number exponent, in a limit process. However, to define and understand a complex number exponent of e, a different type of generalization is required for the concept of exponentiation.

In fact, several definitions are possible. All of them can be proven to be well-defined and equivalent, although the proofs are not included in this article.

Taylor series definition

It is well-known that, for any real x, the following series is equal to ex:

(in other words, this is the Taylor series for the real exponential function, and it has an infinite radius of convergence). This invites the following definition of ez for complex z:

This can be proven to be well-defined; in particular, the series

converges for any z.

Analytic continuation definition

A simple-to-state, equivalent definition is that ez, for complex z, is the analytic continuation of the function ex for real x. This can be proven to be well-defined; in particular, it yields a single-valued function on the complex plane.

Limit definition

It is well-known that, for any real x, the following limit is equal to ex:

This motivates the following definition of ez for complex z:

Differential equation definition

For real x, the function f(x)=ex is well-known to be the unique real function satisfying the differential equation:

for all x. This motivates a definition of f(z)=ez for complex z as the function that satisfies the differential equation:

for all complex z, where the derivative in f'(z) is defined in the sense of a complex derivative. This can be proven to yield a unique function which is well-defined everywhere on the complex plane.

Multiplicative property definition

We would expect the function ez to have the following properties:

e0 = 1 e1 = e

.ez is continuous

ProofIt turns out that this uniquely specifies a function on the complex plane.

Various proofs of this formula are possible. The first proof below starts with the "Taylor series definition" of ez, while the other two use the "Differential equation definition" of ez (see above).

Using Taylor seriesHere is a proof of Euler's formula using Taylor series expansions as well as basic facts about the powers of i:

and so on. The functions ex, cos x and sin x of the (real) variable x can be expressed using their Taylor expansions around zero:

For complex z we define each of these functions by the above series, replacing the real variable x with the complex variable z. This is possible because the radius of convergence of each series is infinite. We then find that

The rearrangement of terms is justified because each series is absolutely convergent. Taking z = x to be a real number gives the original identity as Euler discovered it.

Using calculusDefine the (possibly complex) function ƒ(x), of real variable x, as

The derivative of ƒ(x), according to the product rule, is:

Therefore, ƒ(x) must be a constant function in x. Because ƒ(0) is known, the constant that ƒ(x) equals for all real x is also known. Thus,

Multiplying both sides by eix and using

it follows that Q.U.D.

Using ordinary differential equations

Define the function g(x) by

Considering that i is constant, the first and second derivatives of g(x) are

because i 2 = −1 by definition. From this the following 2nd-order linear ordinary differential equation is constructed:

or

Being a 2nd-order differential equation, there are two linearly independent solutions that satisfy it:

Both cos and sin are real functions in which the 2nd derivative is identical to the negative of that function. Any linear combination of solutions to a homogeneous differential equation is also a solution. Then, in general, the solution to the differential equation is

for any constants A and B. But not all values of these two constants satisfy the known initial conditions for g(x):

.

However these same initial conditions (applied to the general solution) are

resulting in

and, finally,

Q.E.D.

for all and , then is absolutely homogeneous function of degree .

If there exists an , such that

for all and , then is a positively homogeneous function of degree .

Notes

For any homogeneous function as above, .

When the type of homegeneity is clear one simply talks about -homogeneous functions.

! Theorem 1 (Euler)   Let be a smooth homogeneous function of degree . That is,

(*)

Then the following identity holds

Proof. By homogeneity, the relation (*) holds for all . Taking the t-derivative of both sides, we establish that the following identity holds for all :

To obtain the result of the theorem, it suffices to set in the previous formula.

Sometimes the differential operator is called the Euler operator. An equivalent way to state the theorem is to say that homogeneous functions are eigenfunctions of the Euler operator, with the degree of homogeneity as the eigenvalue.