Calculus/Hyperbolic logarithm and angles

After Euler's precalculus, the answer for the derivative of log is unexpected, but when information of quadrature of the hyperbola is supplied, then the answer is as expected. Similarly, when the reciprocal of the variable is integrated, and only Euler's precalculus is supplied, then the answer to the integral appears accidental. Informed of the hyperbolic logarithm as a quadrature, the answer is immediate from the definition of log.

Hyperbolic sectors and dented trapezoids

The top of the trapezoid is dented by curve y=1/x

Standard position hyperbolic sectors have a side through (1,1) and the other side through (t, 1/t) where t > 1.

Consider the triangles T = {(0,0), (1,0), (1,1)} and S = {(0,0), (t,0), (t,1/t)}. Then area T = area S = 1/2. For t > 1 a dented trapezoid is obtained from a hyperbolic sector by adding S and subtracting T.

For 0 < t < 1 a dented trapezoid is obtained from the hyperbolic sector by adding T and subtracting S. The area of the dented trapezoid equals the area of the hyperbolic sector in each case as the same area was added as was subtracted.

Hyperbolic rotations

Area of sectors and rectangles is preserved by hyperbolic rotation

For a positive real number p, a hyperbolic rotation of the plane maps (x,y) to (px, y/p). For any constant c, the hyperbola xy = c is mapped to itself by the hyperbolic rotation, similar to the way the rim of a circular disk is moved, or permuted, by a circular rotation. But a circular rotation preserves distances between points; a hyperbolic rotation preserves areas.

Signed area and directed angles

Circular angle in the range (-π, π) includes negative angles in quadrants III and IV. Suppose area below the x-axis is considered negative, and area above positive. Then the area of a circular sector of a circle of radius √2 with center on the x-axis gives the magnitude of any angle in that range, usually taken with the zero angle pointing right on the axis.

The line y = x is taken to divide the plane into positive and negative halfs for the definition of directed hyperbolic angle. A point (x, y) is in the positive half if x > y. Hyperbolic angle is first defined in quadrant I, with zero angle set by (1,1), and the scope of the angle described by the hyperbola y = 1/x.

Standard position hyperbolic angles have a side through (1,1) and the other side through (p, 1/p) where p is a positive real number. When p > 1, then the magnitude of the angle is positive, while 0<p<1 gives a negative angle as the magnitude is given by sector area, taken to be negative on the upper side of y = x.

The hyperbolic angle in standard position to (p, 1/p) has an area called log p. The area of the corresponding dented trapezoid is thus also log p.

For an arbitrary interval [a, b] in the positive real numbers, there are hyperbolic angles in standard position to (a, 1/a) and (b, 1/b). Subtracting the first from the second gives a hyperbolic angle that corresponds to a dented trapezoid over the interval, and having an area log b – log a.

Furthermore, p = 1/a produces a hyperbolic rotation which moves the hyperbolic angle to standard position with one side on (b/a, a/b). The area of this rotated angle is log (b/a), and since hyperbolic rotation preserves areas, the hyperbolic logarithm has the property log b – log a = log (b/a).

The hyperbolic logarithm and hyperbolic rotation were described by Gregoire de Saint-Vincent in 1647. By exploiting the number p = e = 2.71828... Leonard Euler circumvented the hyperbolic logarithm in 1748 through invocation of the inverse of the exponential function e^x. Following Euler, the function is known as the "natural logarithm".

Application of hyperbolic angle in kinematics in described in a later chapter of this wikibook: Hyperbolic angle.

References

• Areas under a hyperbola and the logarithm from Math for Humans in Future Learn from University of New South Wales – Sydney

• Logarithm as an integral, Mat 21B: Integral Calculus, from University of California, Davis via Libretexts