Transcendental Functions & Hyperbolic Trigonometry

1. Exponential Functions

For a fixed positive base $a > 0$ with $a \ne 1$, the exponential function with base $a$ is defined for all $x \in \mathbb{R}$ by $f(x) = a^x$.

Fundamental Exponential Laws: For all $x, y \in \mathbb{R}$ and $a, b > 0$:

The function is strictly positive: $\operatorname{Range}(a^x) = (0, \infty)$. It is strictly increasing if $a > 1$, and strictly decreasing if $0 < a < 1$.

2. Euler's Number $e$ & The Natural Exponential

The base of the natural exponential function, Euler's number $e \approx 2.718281828459...$, is uniquely defined by the fundamental limit:

Geometrically, $y = e^x$ is the unique exponential curve whose tangent line at the $y$-intercept $(0, 1)$ has a slope of exactly $1$.

3. Logarithmic Functions

Because $f(x) = a^x$ is strictly monotonic on $\mathbb{R}$, it is a bijection from $\mathbb{R}$ to $(0, \infty)$. Its inverse function is the logarithm to base $a$:

The inverse of the natural exponential $e^x$ is the natural logarithm $\ln(x) = \log_e(x)$:

Logarithmic Properties & Change of Base:

1. The Unit Circle Definition of Trigonometric Functions

In analytical calculus, angles are measured strictly in radians. Let $(x, y)$ be the terminal coordinates of an angle $\theta \in \mathbb{R}$ on the unit circle $x^2 + y^2 = 1$ measured counterclockwise from $(1, 0)$:

The reciprocal functions are $\sec\theta = 1/\cos\theta$, $\csc\theta = 1/\sin\theta$, and $\cot\theta = 1/\tan\theta = \cos\theta/\sin\theta$.

2. Fundamental Trigonometric Identities

From the Pythagorean theorem $x^2 + y^2 = 1$ on the unit circle:

Angle Addition & Double-Angle Formulas:

Power-Reduction (Half-Angle) Formulas: Crucial for integration in calculus:

1. Restriction of Domains & Principal Branches

Because trigonometric functions are periodic, they fail the horizontal line test across $\mathbb{R}$. To construct inverses, we restrict each function to a standard principal interval on which it is strictly monotonic and surjective onto its range:

2. Fundamental Identities of Inverse Trigonometric Functions

Cancellation Cautions:

For arguments outside the principal range, symmetry must be utilized: e.g., $\arcsin(\sin(5\pi/6)) = \arcsin(1/2) = \pi/6 \ne 5\pi/6$.

1. Formal Definitions of Hyperbolic Functions

The hyperbolic functions are defined as the symmetric and antisymmetric linear combinations of the exponential functions $e^x$ and $e^{-x}$:

Reciprocals: $\operatorname{sech} x = \frac{1}{\cosh x}$, $\operatorname{csch} x = \frac{1}{\sinh x}$, $\operatorname{coth} x = \frac{\cosh x}{\sinh x}$.

2. The Fundamental Hyperbolic Identity & The Unit Hyperbola

Geometric Analogy: Just as $(\cos t, \sin t)$ parametrizes the unit circle $x^2 + y^2 = 1$, the parametric coordinates $(x, y) = (\cosh t, \sinh t)$ trace the right branch of the unit equilateral hyperbola:

Derived Identities:

1. Invertibility of Hyperbolic Functions

• $\sinh x$ is strictly increasing on $\mathbb{R}$ with range $\mathbb{R}$. Its inverse $\operatorname{arsinh} x$ is defined for all $x \in \mathbb{R}$.
• $\cosh x$ is even on $\mathbb{R}$ with range $[1, \infty)$. Restricting to $x \ge 0$ yields the principal branch of $\operatorname{arcosh} x$ for $x \ge 1$, with range $[0, \infty)$.
• $\tanh x$ is strictly increasing with range $(-1, 1)$. Its inverse $\operatorname{artanh} x$ is defined on $(-1, 1)$.

2. Rigorous Derivation of Logarithmic Closed Forms

Theorem: For all $x \in \mathbb{R}$, $\operatorname{arsinh} x = \ln(x + \sqrt{x^2 + 1})$.

Proof: Let $y = \operatorname{arsinh} x$. Then $x = \sinh y = \frac{e^y - e^{-y}}{2}$. Multiplying by $2e^y$ yields:

This is a quadratic equation in $u = e^y$. By the quadratic formula:

Since $e^y > 0$ for all real $y$, and $\sqrt{x^2+1} > \sqrt{x^2} = |x| \ge x$, the minus sign yields a strictly negative value $x - \sqrt{x^2+1} < 0$, which is inadmissible. Thus:

Complete Logarithmic Catalog: