Gudermannian function

pahio

2013-03-22 3:14:48

The Gudermannian function gd is defined by the definite integral

gdx :=

x0

dt

cosh t=

x0

2 dt

et + et= 2

x0

et

e2t + 1dt. (1)

Because of the continuity of the integrand of (1), this equation defines a differ-entiable real function, which by the fundamental theorem of calculus has thederivative

d

dxgdx =

1

coshx. (2)

From this we infer that the Gudermannian is an odd function.Using the substitution et = u we can make from (1) the closed form

gdx = 2 arctan ex pi2. (3)

This enables to see that

limx gdx =

pi

2, lim

x+ gdx = +pi

2,

whence the graph of the gudermannian has the lines y = pi2 as asymptotes.Besides (3), one has also e.g. the closed forms

gdx = arcsin(tanhx), gdx = arctan(sinhx),

since both of these composition functions have the derivative1

coshx, as (2),

and vanish in the origin (see the fundamental theorem of integral calculus).Accordingly, we may write the formulae

sin(gdx) = tanhx, tan(gdx) = sinhx (4)

which are illustrated by the below right triangle. Cf. the properties of thecatenary!

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1

sinhxcoshx

1

gdx.

.

Thus, Gudermannian function offers a beautiful connection between thetrigonometric and the hyperbolic functions without using imaginary numbers(cf. the paragraph 3 in hyperbolic identities).

The Taylor series expansion of gdx may be expressed with the Euler numbersEn; these have as generating function the derivative of Gudermannian function:

1

coshx:=

n=0

Enn!

xn (|x| < pi2

)

gdx =n=0

En(n+1)!

xn+1 (|x| < pi2

) (5)

Since gd is odd (and its derivative even), the numbers En with odd indices are0; the others are non-zero integers. The begin of (5) is

gdx = x 13!x3 +

5

5!x5 61

7!x7 +

1385

9!x9 + . . .

The curve y = gdx

1 2 3 4 512345

1

2

1

x

y

.

2