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ANALYSIS II
Special Functions

Defn The logarithm function is defined by

Properties

• L(1) = 0, L'(x) = 1/x, L strictly increasing and continuous.
  
• L(x₁ x₂ ) = L(x₁) + L(x₂)
  Pf: Use additivity of integration over subintervals, and change of variables.
  
  Corollaries
• L(xⁿ ) = n L(x), lim _x   L(x) =
  Pf: Use induction with the previous result.
  
• L(1/x) = - L(x), lim _{x  0} L(x) = -
  Pf: Notice that L(x 1/x) =0.
  
• Dom(L) = R⁺, Range(L) = R.
  Pf: Since L is continuous, by the intermediate value theorem each real number must be in the range.
  

Defn The real number e is defined as the unique number such that L(e) = 1.

This number is unique since L is 1:1 ( which follows since L is strictly increasing).

• e >2
  Pf: Use the fact that L strictly increases, L(e) = 1 > L(2).
  
• (L')'(x) = -1/x² < 0 ; L' decreases and L is concave down.
  

Defn  The logarithm function with base a>0 is defined by log_a(x) = L(x)/L(a).

This function satisfies all the properties of logarithm above, but with log_a(a) = 1..

Defn  The exponential function is defined as the inverse function of L(x), i.e. E(x) = L^-1(x).

Properties

• E(0) = 1, E(x) is a strictly increasing function with domain R and range R⁺.
  
• E is continuous and differentiable with E'(x) = E(x).
  Pf: Use the formula to compute derivatives of inverse functions: g'(x) = 1/(f'(g(x)), if g=f^-1.
  
• E(x₁ + x₂ ) = E(x₁) E(x₂),  E(mx) = E(x)^m.
  Pf: Use the fact that    E(x) = y    if and only if    L(y) = x.       Set y_j = E(x_j ), j=1,2.
  
• E(1/n) = e^(1/n), E(m/n) = e^(m/n)  (for each rational m/n).
  Note: This allows us to extend e^x to all real numbers as E(x) by using the fact that the rationals are dense and E is continuous.
  Pf: Let y = E(1/n), then yⁿ =.E(n 1/n) = e. So y is the n-th root of e. The second statement follows as before by induction.
  
• (E')'(x) = E(x)>0 , so E' strictly increases and E is concave up.

Defn The exponential function a^x = E(x L(a)) has all the anticipated properties and is the inverse function of log_a (x).