Good Things about the Gudermannian. A Series Illustrating Innovative Forms of the Organization & Exposition of Mathematics by Walter Gottschalk

Transcription

1 Good Things about the Gudermannian #88 of Gottschalk s Gestalts A Series Illustrating Innovative Forms of the Organization & Eosition of Mathematics by Walter Gottschalk Infinite Vistas Press PVD RI 003 GG88 (31)

2 003 Walter Gottschalk 500 Angell St #414 Providence RI 0906 ermission is granted without charge to reroduce & distribute this item at cost for educational uroses; attribution requested; no warranty of infallibility is osited GG88-

3 , y, u, t Œreal nr var the real gudermannian function the gudermannian ab goohd - er - MAHN - ee - en r gd (- < < ) dn wh gd gudermannian goohd - er rd Tan sinh df GG88-3

4 the real gudermannian constitutes a real bridge between the real trigonometric functions and the real hyerbolic functions viz y gd fi sin y tanh cos y sech tan y sinh cot y csch sec y cosh csc y coth & tan y tanh cot y coth GG88-4

5 the corresondence between the trig fcns and the hy fcns via the gudermannian roduces a corresondence between trig identities and hy identities; eg looking at the three trig ythagorean identities trig: sin + cos 1 hy: tanh + sec h 1 trig: 1 + tan sec hy: 1 + sinh cosh trig: 1 + cot csc hy: 1 + csc h coth GG88-5

6 the coresondence between trig fcns & hy fcns via the gudermannian has a geometric descrition which is given by the following three labeled right triangles GG88-6

7 1 sin gd tanh gd cos gd coth sec gd cosh tan gd sinh gd 1 csc gd coth 1 gd cot gd csch GG88-7

8 forms of the gudermannian gd sin Ú 0 tanh cos sec h tan sinh cot csc h sec cosh csc coth tan tanh tan e - sech t dt ( domains & ranges have to be secified) GG88-8

9 forms of the inverse gudermannian gd sinh cosh tanh coth sech csch tanh tan Ê log tan + ˆ Ë 4 log (sec + tan ) Ú sec t dt 0 tan sec sin csc cos cot ( domains & ranges have to be secified) GG88-9

10 roerties of the gudermannian y gd and its grah D the function y gd has these roerties domain: - < < range: - < y < class: analytic arity: odd strictly increasing gd ( 0) 0 gd > 0 > 0 gd < 0 < 0 $ lim gd Æ $ lim gd - Æ- GG880

11 D the grah of y gd has these roerties thru origin with sloe 1 symmetric wrt origin steadily rising asymtotic to horizontal line y asymtotic to horizontal line y fle oint at origin concave down for > 0 concave u for < 0 - GG881

12 arametric equations, first form ÏÔ tanh Ì - ÓÔ y tan wh t < 1 1 t t arametric equations, second form ÏÔ coth Ì - ÓÔ y cot wh t > 1 1 t t note: the values of the inv trig fcns tan t & cot are to be roerly chosen t GG88

13 do - it - yourself sketch: grah of the gudermannian y gd (- < < ) y - GG883

14 roerties of the inverse gudermannian y gd and its grah D the function y gd domain: - < < range: - < y < class: analytic arity: odd strictly increasing gd ( 0) 0 gd > 0 > 0 gd < 0 < 0 gd Æ as gd Æ - as Ø - has these roerties GG884

15 D the grah of y gd ha s these roerties thru origin with sloe 1 symmetric wrt origin steadily rising asymtotic to vertical line asymtotic to vertical line fle oint at origin concave u for > 0 concave down for < 0 - GG885

16 arametric equations, first form ÏÔ tan Ì ÓÔ y tanh wh t < 1 t t arametric equations, second form ÏÔ cot Ì ÓÔ y coth wh t > 1 t t note: the values of the inv trig fcns tan t & cot are to be roerly chosen t GG886

17 do - it - yourself sketch: grah of the inverse gudermannian y gd Ê - < < ˆ Ë y - GG887

18 derivatives and differentials d gd sech d d gd d sec dgd sech d dgd sec d GG888

19 indefinite and definite integrals Ú sech d gd +C Ú sec d gd +C Ú sech t dt gd 0 Ú sec t dt gd 0 GG889

20 the Maclaurin series for gd gd  En n + 1 ( n + 1)! n 0 IC : - 1 < < 1 GG88-0

21 the Maclaurin series for gd gd  n 0 ( 1) n En ( n + 1)! n + 1 IC : - < < GG88-1

22 more formulas tanh 1 tan 1 gd & coth 1 cot 1 gd e sec gd + tan gd 1+sin gd cos gd cos gd 1 - sin gd tan Ê 1 gd + Ë ˆ 4 gd Ê + ˆ log (csc - cot ) Ë d gd d Ê Ë + ˆ csc Ú csc d gd Ê + ˆ Ë + C GG88-

23 the si basic trig functions may be rationalized by the substitution utan viz sin u 1 u cos u 1 + u tan u cot 1 - u 1 - u u csc 1+u u sec u u & du d + 1 u so that some trig integrands may be rationalized by this substitution GG88-3

24 a geometric descrition of the substitution utan is given by the labeled right triangle 1 + u u 1 - u GG88-4

25 the si basic hy functions may be rationalized by the substitution utanh viz sinh u 1 u cosh u 1 - u tanh u 1 + u coth 1 + u u 1 u sech - csch 1 - u 1 + u u & du d - 1 u so that some hy integrands may be rationalized by this substitution GG88-5

26 a geometric descrition of the substitution utanh is given by the labeled right triangle 1 + u u gd 1 - u GG88-6

27 a definite integral the area of the region in QI bounded by the curve y gd & the horizontal line y & the y - ais the area of the region in QI bounded by the curve y gd & & the vertical line the - ais GG88-7

28 Ê ˆ Ú -gd d 0 Ë Ú 0 Ú 4 gd d log cot d 0 c/ ª wh c/ cl Catalan' s constant df  n 1 () ( n + 1) n G88-8

29 further uses of the gudermannian o u in various laces such as the theory of ellitic functions, noneuclidean geometry, hysics of the endulum, and cartograhy; indeed in the Mercator ma rojection the vertical distance from the equator of a location on the chart is given by gd J where J is the latitude of the location GG88-9

30 bioline Christoh Gudermann German analyst, geometer; teacher of Weierstrass; name ' gudermannian' and notation ' gd' in resent usage were introduced by Cayley in honor of Gudermann's work in the area GG88-30

31 IMHO the gudermannian should aear in several eercises scattered thruout the undergraduate calculus courses; for eamle, it hels to clarify that mysterious formula for the indefinite integral of the secant and indeed rovides a short formula for it; likewise for the cosecant; it correlates the trig functions and the hyerbolic functions in a leasant and surrising manner GG88-31