A Series Illustrating Innovative Forms of the Organization & Exposition of Mathematics by Walter Gottschalk

Transcription

1 The Inverse Function Tableau #2 of Gottschalk's Gestalts A Series Illustrating Innovative Forms of the Organization & Exposition of Mathematics by Walter Gottschalk Infinite Vistas Press PVD RI 2000 GG2-1 (29)

2 2000 Walter Gottschalk 500 Angell St #414 Providence RI permission is granted without charge to reproduce & distribute this item at cost for educational purposes; attribution requested; no warranty of infallibility is posited GG2-2

3 the inverse function tableau the inverse function tableau = IFT is a simple arrangement of text that displays the conceptual determination of the inverse function of a given function or the end result of a computational determination of the inverse function of a given function; if the inverse function is multiple - valued, then a principal - valued inverse function may be included; IFT is, in particular, a convenient device to indicate the notation, domains, and ranges of the functions involved GG2-3

4 to construct the IFT of a function y=f(x), read ' y equals f of x', solve for x in terms of y (where ' solving' may mean more of a thought than an algorithm) to obtain an equivalent equation x = and then interchange x and y to make x the independent variable and y the dependent variable, thus obtaining the inverse function -1 f - 1 y=f (x), read ' y equals f inverse of x', with the original notation for independent and dependent variables; (y), GG2-4

5 this procedure may be indicated schematically as follows: -1 y= fx ( ) x = f ( y) -1 x= f( y) y = f ( x) where the mutually inverse functions, with the same notation for independent and dependent variables, appear in the upper left corner and the lower right corner; if the inverse function is multiple - valued, then functional values may be chosen to form a principle - valued inverse function which is a single - valued function with the same domain; note that the inverse function of the inverse function is the original function ie the inverse function of y = f (x) is y = f( x) ; -1 results are arranged in a certain way for oversight and insight; various examples are given below GG2-5

6 notation and geometric names for the five basic number systems of classical analysis (0) R, R, R, C, C (1) the real line = the set of all real numbers = df dn = (open cap) ar rd R ( 2) the extended real line = = df dn = (open cap) ar bar rd {- }» R» { + } R (3) the projective real line = df R» { } = dn R = (open cap) ar dot rd GG2-6

7 (4) the complex plane = the set of all complex numbers = df dn = ( open cap) cee rd C (5) the complex sphere = df C» { } = dn C = ( open cap) cee dot rd GG2-7

8 some convenient abbreviations function = fcn domain = dmn range = rng inverse = inv self - inverse = si single - valued = sv double - valued = dv multiple - valued = mv infinitely - many - valued = v principal - valued = pv GG2-8

9 variable = var real number variable = real variable = real var complex number variable = complex variable = complex var variable which ranges over items = item variable = item var the set of all variables whose range is the set S = var S the downward arrow ' has a / an / the' may be read GG2-9

10 E1. the general real linear function IFT y = ax + b (x ŒR) (a,b ŒR & a π 0) (x, y Œreal var) fcn y=ax+b dmn: - < x < + rng: - < y < + sv inv fcn 1 y = - - a x b a dmn: - < x < + rng: - < y < + GG2-10

11 E2. the general complex linear function IFT w = az + b (z ŒC) ( ab, ŒC &aπ 0) ( zw, Œcomplex var) fcn w = az +b dmn: z ŒC rng: w ŒC sv inv fcn 1 w = - a z - dmn: z ŒC rng: w ŒC b a GG2-11

12 E3. the real square function IFT y = x ( x ŒR) 2 (x, y Œreal var) fcn y = x dmn: - < x < + rng: 0 y < + dv inv fcn y = ± x dmn: 0 x < + rng: - < y < + pv inv fcn y= x dmn: 2 0 x < + rng: 0 y < + GG2-12

13 E4. the complex square function IFT w = z ( z ŒC) ( zw, Œcomplex var) 2 fcn 2 w = z dmn: z ŒC rng: w ŒC dv inv fcn w = ± z dmn: z ŒC rng: w ŒC pv inv fcn w = z dmn: z ŒC rng: 0 Ang w < p GG2-13

14 E5. the real exponential function IFT y = e (x ŒR) x (x,y Œreal var) fcn y = e x dmn: - < x < + rng: 0 < y < + sv inv fcn y = log x = ln x dmn: 0 < x < + rng: - < y < + GG2-14

15 E6. the complex exponential function IFT w= e (z ŒC) (z, w Œcomplex var) z fcn z w= e dmn: z ŒC rng: w ŒC & w π 0 v inv fcn w= log z dmn: z ŒC & z π 0 rng: w ŒC pv inv fcn w = Log z dmn: z ŒC & z π 0 rng: w Œ C & 0 Imw < 2p for 0 π z ŒC: log z= ln z + i ang z = ln z + i( Ang z + 2pZ) Log z = ln z + iang z wh 0 Ang z < 2p GG2-15

16 E7. the real sine function IFT y = sin x (x ŒR) ( xy, Œreal var) fcn y = sin x dmn: - < x < + rng: -1 y 1 v inv fcn y = sin -1 dmn: - 1 x 1 rng: - < y < + pv inv fcn -1 x y = Sin x dmn: - 1 x 1 p p rng: - y 2 2 GG2-16

17 E8. the complex sine function IFT w = sin z ( z ŒC) (z, w Œcomplex var) fcn w = sin z dmn: z ŒC rng: w ŒC v inv fcn w = sin -1 z dmn: z ŒC rng: w ŒC pv inv fcn -1 1 w = Sin z = Log ( iz + - ) i z 2 1 dmn: z ŒC rng: set of all w as above GG2-17

18 E9. some self - inverse functions the real negation function f(x) = - x on the real line or on the extended real line or on the projective real line the complex negation function f(z) = - z on the complex plane or on the complex sphere GG2-18

19 the real reciprocal function 1 f(x) = x on the punctured real line or on the projective real line the complex reciprocal function 1 f(z) = z on the punctured complex plane or on the complex sphere the complex conjugation function f(z) = z on the complex plane or on the complex sphere any combination of negation, reciprocation, conjugation GG2-19

20 transposition of matrices conjugation of matrices conjugate transposition of matrices inversion of matrices formation of dual spaces of vector spaces inversion in any group negation in any additive group reciprocation in any field GG2-20

21 complementation of sets conversion of relations passage to the dual in any duality theory passage to the converse in logic passage to the contrapositive in logic interchange of two specified items in an array reflection in a point / line / plane / etc inversion of functions ie the inverse function of the inverse function is the original function etc GG2-21

22 E10. the real Joukowski transformation 1 1 IFT y = ( x + ) (x Œ ) 2 x (x, y Œvar R ) fcn 1 1 y = ( x + ) 2 x dmn: x ŒR rng: - < y y < + y = dv inv fcn y = x ± x -1 dmn: - < x x < + x = rng: y ŒR pv inv fcn y= x + x dmn: - < x x < + x = rng: - 1 y 0 1 y < + y = GG2-22

23 E11. the complex Joukowski transformation 1 1 IFT w = ( z + ) (z ŒC ) 2 z (z, w Œvar C ) fcn 1 1 w= ( z + ) 2 z dmn: z ŒC rng: w ŒC dv inv fcn 2 w = z ± z -1 dmn: z ŒC rng: w ŒC GG2-23

24 E12. the general real homography ax + b IFT y = (x ŒR ) cx + d (a, bcd,, ŒR& ad -bcπ 0) ( xy, Œvar R ) fcn ax + b y = cx + d dmn: x ŒR rng: y ŒR sv inv fcn dx - b y= - cx + a dmn: x ŒR rng: y ŒR GG2-24

25 E13. the general complex homography IFT w = az + b (z ŒC ) cz + d ( abcd,,, ŒC& ad -bcπ 0) ( zw, Œvar C ) fcn w = az + b cz + d dmn: z ŒC rng: w ŒC sv inv fcn dz - b w= - cz + a dmn: z ŒC rng: w ŒC GG2-25

26 E14. the gudermannian IFT y= gdx (x ŒR) (x, y Œreal var) fcn -1 y= gdx = Tan sinh x = secht dt dmn: - < x < + p p rng: - < y < 2 2 sv inv fcn -1-1 y= gd x = sinh tan x = sec t dt = ln(sec x + tan x) p p dmn: - < x < 2 2 rng: - < y < + Ú x 0 Ú x 0 GG2-26

27 was ist gut about the gudermannian? one good thing (there are others) is that it provides a real bridge between the trigonometric functions & the hyperbolic functions viz u= gd x fi sin u = tanh x cos u = sech x tan u = sinh x cot u = csch x sec u = cosh x csc u = coth x GG2-27

28 E15. two especially notable examples of inversion The Fundamental Theorem of Calculus states that differentiation & integration are inverse operations; what one does, the other undoes elliptic functions & elliptic integrals are inverse functions of each other (with lots of technical details) GG2-28

29 historical note inspired by the example of the inverse elliptic functions & elliptic integrals & attendant insights following from the recognition of this inverse relationship, Jacobi said Du muss immer umkehren! (German) = Thou must always invert! thus proclaiming that unremitting inversion is the secret of success in mathematics GG2-29