Views
3 years ago

1. Functional equations in one variable

1. Functional equations in one variable

If xf ( x ) < 1, then by

If xf ( x ) < 1, then by (ii) we have condition 2. Thus, we must have Remark. 1 xf( x) xf ( x ) = 1. This means Mathematical Database is a fixed point larger than 1. This again contradicts with f( x) 1 = . x The concept of fixed point is introduced in this question. If f ( x) of f. It is not so commonly used but it is also a useful technique. = x, then x is called a fixed point 3. Some famous functional equations In this part, we will introduce some famous functional equations. We may quote them directly in competitions. Theorem 3.1. (Cauchy equation) If f is a continuous function such that f ( x+ y) = f( x) + f( y) for all xy∈ , , then f ( x) = cx where c is a constant. Proof. We may use the technique mentioned in example 2.1. First, put f ( x) y = 1 and let x be a positive integer. Then f ( x+ 1) = f ( x) + c, where c = f(1) . Thus = cx for positive integers x. It is easy to verify that (0) 0 original equation. For negative integers, we replace x by –x and get f = by putting x = y = 0 into the f ( − x+ 1) = f( − x) + c. Thus f ( − x) =−cx for positive integers x. Therefore we conclude that f ( x) = cx for integers x. Let m x = , where m, n are integers. Then we have n ⎛m+ 1⎞ ⎛m⎞ ⎛1⎞ f ⎜ ⎟= f ⎜ ⎟+ f ⎜ ⎟ ⎝ n ⎠ ⎝ n ⎠ ⎝n⎠ , thus ⎛m ⎞ ⎛1 ⎞ f ⎜ ⎟= mf ⎜ ⎟ ⎝ n ⎠ ⎝n⎠ . However, 1 ⎛1⎞ c = f ( ⋅ n ) = nf ⎜ ⎟ n ⎝n⎠ ⎛1 ⎞ c f ⎜ ⎟ = ⎝n⎠ n and we have ⎛1 ⎞ c f ⎜ ⎟ = ⎝n⎠ n . This implies f ( x) = cx for rational numbers x. . So Page 8 of 10

Mathematical Database Now, as f is continuous, we can always bound an irrational number by two rational numbers. (For example, we may bound π by 3, 3.1, 3.14, 3.141 and 4, 3.2, 3.15, 3.142 respectively). Let { xn}, xn∈ be such a sequence with lim x n = x , then by continuity of f, we have n→∞ f ( x) = f(lim x ) = lim f( x ) = lim cx = cx. Therefore f ( x) n n n n→∞ n→∞ n→∞ = cx for all x∈R . Q.E.D. Corollary 3.2. If f is a continuous function and for all xy∈ , , (i) f ( x+ y) = f( x) f( y) then f ( x) = c (ii) f ( xy) = f ( x) + f ( y) then f ( x) = clnx (iii) f ( xy) = f ( x) f ( y) then f ( x) = x where c is a constant. x c Example 3.3. If f :(1, +∞) → is a continuous function such that f ( xy) = xf ( y) + yf ( x) for all1 < xy , ∈ , find f ( x ). Solution. It is equivalent to f ( xy) = f ( x) + f ( y) . If we let xy x y gxy ( ) = gx ( ) + gy ( ) f ( x) gx ( ) = , then the equation becomes x which is simply (ii) of corollary 1.32 (although there is a minor difference between them). So we have g( x) = clnxand f ( x) = xg( x) = cxlnx. Page 9 of 10

3.1-Linear Equations in 2 variables
Linear algebra c-4 - Quadratic equations in two or three variables
Real Functions of Several Variables - Nabla...
3-5 Solving Equations with the Variable on Each Side
How to work on algebra equation
Judgments of Functional Relationships in Systems of Three Variables
Renormdynamic equations and scaling functions of multi ... - JINR
3-1 Writing Equations 3-1 Writing Equations - Mona Shores Blogs
Radial Basis Functions for Computational Geosciences*
Likelihood and Estimating Equation-Based Variable ... - SAMSI
Dirac's Equation and the Sea of Negative Energy (Part 1) - Open SETI
2008, VOLUME 1 - The World of Mathematical Equations
Thomson ONE Banker Tip Sheet PFDL functions
Early Prediction of Type 1 Diabetes with Functional Genomics
Chapter 1: Equations and Inequalities - Unit 5
Elementary Analytic Functions - Complex Functions Theory a-1
1.Feature and Function - Anydiag.com
Natewa Tunuloa Peninsula is 1 of 14 Important ... - Equator Initiative
“Serving society is only one of higher education's functions, but ...
PRIMUS: Evolution of the Stellar Mass Function Since z=1
Supplement 1 - TIMSS and PIRLS Home - Boston College
Supplement 1 - TIMSS and PIRLS Home - Boston College
Solutions of Equations in One Variable
IMO Problems on Functional Equation
Solutions of equations in one variable
Linear equations in one variable - Math Centre
Chapter 2 Solutions of Equations in One Variable
Functional Equations - The World of Mathematical Equations