Table of a Weierstrass Continuous Non-Differentia ble Function

Table of a Weierstrass ContinuousNon-Differentia ble Function

By Herbert E. Salzer and Norman Levine

Many studies have been made of continuous non-differentiable functions [1], the

most famous of which is Weierstrass's W(a, b, x) defined by

00

(1 ) W(a, b, x) = X a" cos t¿»"**), 0 < a < 1, b an odd integer.7.-1

It is shown in some books [1], [2] that for

(2) ab>l+*£,

W(a, b, x) is continuous everywhere and has no derivative anywhere, but Bromwich

[3] improved this condition to

(3) nlt> 1 +$0 - «),

which, according to Hardy [4] is the sharpest result (as of 1916) for no derivative,

finite or infinite. (Hardy showed b > 1, ab ^ 1 sufficient to establish the non-exist-

ence of any finite derivative. He also showed that those same conditions, together

with a(b + 1) < 2 for b - ik + 1, permitted the existence of an infinite derivative

at certain points.) To illustrate the difference between (2) and (3) for a = §,

(2) requires b ^ 13, while (3) permits b = 7. However, as far as the authors know

there may be considerable work to lie done in the direction of lowering the boundo

of I -}- -,-- ( 1 — a) in (3) for the case of no derivative, finite or infinite.

Owing to the unusual nature of W(a, b, x) and the absence of any previous table,

or even graph, despite the countless number of theoretical papers, it was believed

that an extensive table of this Weierstrass function for some typical pair of param-

eters a and b might be of value as more than a mere curiosity, namely for suggest-

ing or motivating further research, and for its interest to workers in numerical

analysis. Thus, in this last connection, it might !>e of interest to determine empiri-

cally what results in numerical integration and possibly interpolation are available

from the continuity alone. That W(a, b, x) is integrable follows from its continuity,

and one might l>e curious to see the results of applying standard numerical integra-

tion formulas where the usual derivative formulas for the remainder would l>e

inapplicable. Likewise, one might be curious to test out standard Lagrangian inter-

polation, where the remainder is often expressed in terms of derivatives. (We can

write down interpolation and numerical integration formulas, avoiding derivatives

in the remainder terms by employing divided differences and integrals with divided

differences in the integrand, respectively. However, one usually estimates divided

differences in terms of derivatives.) Finally, one's curiosity might extend as far as

Received February 23, 1900; revised July 28, I960.

120

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

TABLE OF A WEIERSTRASS CONTINUOUS NON-DIFFERENTIABLE FUNCTION 121

glancing at the results of standard numerical differentiation and interpretation of

the results in the light of the knowledge that W(a, b, x) has no derivative.

For tabulation of any W(a, b, x), it is immediately apparent from (1) that

(4) W(a, b, 1 + x) -— W(a, b, x),

so that the range of x need not go outside (0, 1). From (I),

^(«,6,0) - ~W(a, b, 1) m a/(\ - a);(5)

W(a, b, J) = 0.

From the trigonometric identity

(6) cos(tot(J ± t)) = T( — 1 )l"-,)/* sin to**, to odd,

we have

(7) W(a,b,k + t) = -W(a,b,h-t),

so that for complete tabulation of any W(a, b, x) it suffices for x to range from

Oto iIn connection with the choice of a and b, it is apparent that for a close to 1, we

can choose b as low as 3, but the convergence of the series in ( 1 ) would be too slow

for practical calculation of W(a, b, x) to high accuracy. Making a very small would

give rapid convergence, but for accuracy fixed at a certain number of decimal

places as a tends to get very small, say

a = (, b" > N = il + ~ (1 - «)}'A"

becomes enormous and W(t, b, x) becomes essentially the first term of (1), « cos

(6"tx), whose graph would appear like that of a very highly oscillatory function of

small amplitude. As a compromise Ijetween these two extreme types, we took

a = \ and b = 7. The choice a = § did not lead to too many terms of ( 1 ), 50 terms

giving a truncating error < £• 10"15, and yet there were sufficient terms beyond the

first few to give a graph that is characteristic of W(a, b, x) rather than a predomi-

nantly sinusoidal type of curve. The 6 = 7 barely satisfies (3), thus tending to

minimize the oscillatory behavior of W(a, b, x) and to facilitate graphing. We shall

denote W(a, b, x) which is tabulated here for a = | and b = 7 by W(x).

This present table of W(x), x - 0( .001)1 to 12D, was printed out and rounded

from a preliminary calculation on the IBM 704 to several more places. Two sepa-

rate and independent print-outs, supposedly identical, were proofread against each

other, with just a single print-out error turning up. Naturally, no differencing check

could be made upon the correctness of this table of TT(x), but every value under-

went the following final functional check:

(8) W(7x) = 2W(x) - cos (7tx),

which was performed by desk calculation upon W(x) on one of the preliminary

print-outs. The results showed W(x) to be correct to around 14D. In employing (8),

W(7x) was found in the table as ± W(xl) for some suitable f, 0 g i* á i accord-

ing to (4) and (7), and cos (7tx), after reduction of 7tx to the first quadrant, was


0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0251.0 *—i-1-1-1-1-1-1-1-1-1-1-[-*-!-1-1-1-i-1-1-!—1-1-1-1-1 1.0

DOO 001 002 003 004 0 05 0 06 0 07 0 08 0.09 0.10 0.11 0.12 0 13 0.14 0 15 0.16 0.17 0.18 0.19 0.20 0.21 0 22 0.23 0 24 0 25

0 24 0 26 0 27 0 28 0 29 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0 37 0.38 0.39 0.40 0.41 0.42 0.43 0 44 0.45 0.46 0 47 0 48 0 49 0 50

0 25 0.26 0.27 0.28 0.29 0 30 0.31 0.32 0.33 0 34 0.35 0 36 0 37 0 38 0.39 0 40 0.41 0 4? 0 43 0 44 0 45 0.46 0 47 0.48 049 0 50

Fig. 1.—illustration of a Weierstrass, Everywhere-Continuous Nowhere-DirTerentiableCO

Function, W(x) = £a" cos (b"wx) a = -|; 6 = 7; x = 0(0.001)0.500n=l

122



looked up in a well-known 15-place table at intervals of 0.01° [5]. The final 12-deci-

mal table was checked by reading it several times against one of the print-outs,

and it is believed to be correct to well within a unit in the 12th decimal.

The purpose of the accompanying figure, which is merely a broken line graph of

the table of W(x), is to furnish at a glance a view of the peculiar behavior of W(x).

Of course, the graphical picture would be more complete if the time and means were

available for calculating W(a, b, x) as a function of a also, and for a sequence of

permissible odd integral values of i> (according to (3)) to correspond to each o.

Although no offhand justification could be found for drawing anything smoother

than a broken line connecting these 500 points, one still finds its ripples of irregu-

larity, superposed upon a broader pattern of smoothness, to be quite revealing as

to the nature of W(x) and how it might appear under repeated "magnification"

(i.e., subtabulation).

To establish (8), replace x by 7x, in W(x) = £"_i cos (7"irx)/2n, to get

TV(7x) = 2 22 cos (7B+Vx)/2B+I = 2 £ cosi7"'«)/2"' = 2W(x) - cos(7ttx).

By repeated application of (8),

W(7nx) - 2W(7"'1x) - cos (7"tx) = éW(7n~2x) - 2 cos (7n_Vx) - cos (7\x)

= 8W(7n~ix) - 4 cos (7b-îtX) - • • • etc. until we reach

n-l

(9) W(7"x) = 2"W(x) - £ 2" cos (7"~r xx).

From (9), for x = 1/7", W(l) = - 1 = 2nW(l/7n) - £ 2r cos (r/T), from

which

(10) W(l/7n) = - 1/2"-1 + "¿ cos (T/T)/2n-T.r-l

Letting n —* *> in (10), we see at once that

(11) lim (2 2r cos (x/T)\ /2n = 1.

To test the value of standard numerical integration formulas upon W(x), whose

integral is given by

(12) f W(t) dt = - ¿ sin (7Birx)/14",

W(t)dt, I W(t)dt, ■ ■ ■ , f W(t)dt were computed analytically•<0.1 •'0.4

from (12), and then were computed numerically by both trapezoidal and Simpson's

rules at intervals of 0.001, with the following results:

Interval

0 to 0.10.1 to 0.20.2 to 0.30.3 to 0.40.4 to 0.5

True Value

0.01899 29-0.04145 650.03084 620.00337 70

-0.03298 02

Trapezoidal Rule

0.01898 76-0.04143 80

0.03084 430.00342 54

-0.03300 67

Deviation

-0.00000 53

+0.00001 85-0.00000 19

+0.00004 84-0.00002 65

Simpson's Rule

0.01901 44-0.04145 43

0.03085 140.00340 27

-0.03288 27

Deviation

+0.00002 15+0.00000 22+0.00000 52+0.00002 57+0.00009 75


124 HERBEBT E. SALZER AND NORMAN LEVINE

The results show no recognizable advantage in Simpson's rule. In fact, the sum of

the absolute values of the above deviations in the trapezoidal rule is around 10~\

while the sum of the absolute values of the Simpson deviations is around 1£-10~"\

This may indicate that no higher-point formula will improve over the trapezoidal

formula.

Lagrangian polynomial interpolation at intervals of 0.002 was tried for the 2-

through 7-point cases, for a mid-interval (i.e., already tabulated) value of W(x) at

two different places, x = 0.007 and x = 0.037, where the true value to 5D is 0.60807

and 0.43362 respectively. At each place the error in almost all cases ranged from

around 0.01 to 0.05. More specifically, for x = 0.007 the error fluctuated between

0.01 for every even-point interpolation and 0.014 to 0.049 for various odd-point

interpolations, and for x ■ 0.037 there were deviations of 0.032 and 0.055 for respec-

tive 2-point and 3-point interpolation and deviations ranging from 0.001 to 0.021

in the higher-point interpolation. On the basis of these two tests alone it would

appear that one could not really count upon any systematic improvement beyond

linear interpolation.

Finally, out of pure curiosity, 2- through 7-point Lagrangian differentiation, for

the "first derivative," was tried out at the tabular interval of 0.001, for x = 0.002,

and surprisingly enough, outside of the 2-point answer of —74 and the 3-point

answer of —133, the remaining four cases all came within 6 units of —150.

From a casual look at the graph of W(x), it is apparent that in place of the de-

rivative there is a general directional trend from any point xo if we do not go too

far away from x0, and we might seek a suitable quantitative estimate for an "aver-

age slope" between x0 and x0 +-h. (The discussion here is concerned with a suitable

generalization of the left- or right-hand derivative, rather than the derivative.)

One suggestion that would appear natural for W(x, a, b), or any other continuous

function, would be to investigate the possibilities of the average of the difference

quotient {/(x) — /(x0))/(x — x0), which exists and is itself continuous for every

x except xo in the open interval (x0, x0 + A). This average difference quotient or

Da/(xo) might have the following definition (assuming that it exists in the first

place):

(13) 2uA/(xo) = ± ] {[f(x) - /(xo)]/(x - xo)} dx.

That (13) may be a suitable generalization follows from the fact that when/'(x0)

exists, (13) exists, and

(14) limaV(xo) = /'(x.).A-»0

This is seen at once from the replacement of [f(x) — /(x0)}/(x — xo) byZ'(xo) +

e (x) in (13) and the continuity of e (x) in the closed set (x0, x0 + h) which makes

e (x) integrable. Thus (13) exists and

(•X0+*

(x)dx 0 as h -* 0,

which implies (14).

It is not difficult to find examples of continuous functions/(x) where f'(x0) does

not exist and (a) also £>a/(x0) does not exist, or (b) £>hf(xo) exists but Iinu_0

3Da/(xo) does not exist. But we may also have (c) no /'(x0), with both 3öa/(x0)

and liñudo ï>kf(xn) existing. In other words the existence of lim*.^ 35»/(xo) still



does not imply the existence of f'(x0). Such a counter-example,* which is due to

the referee, is the following. Let x0 = 0 and

f(x) = x sin - (x t¿ 0)X

/(o) - o.This continuous function has no derivative at x = 0, but

lim £>*/(0) = 0.A-0

First

^ = 1hlhgin(x)dx

exists since the integrand is bounded and continuous except at one point. This

suffices. To estimate 3Dk we let

»/<,+». ain\x)dx = lnr ^ydy-

By the mean value theorem

/„ = (-ir-2/0/where

nx < 0„ < (n + l)x.

Suppose that h = \/(n + a)x, 0 g a < 1. Then

$>h = (n + a)x / y~2 sin y dy + In+i + h+i + • • • ,\_J(n+a)T J

and therefore | 3}A | < (n + a)x | /„ | < 2(w + a)x/nx2.

Therefore as h —■ 0, SD* also —> 0.

The authors wish to acknowledge the assistance of Mrs. Charlene M. Janos in

checking the entire table of W(x) by the functional check (8).

Convair-Astronautics

San Diego, California

1. A. N. Singh, The Theory and Construction of Xon-Differentiable Functions, LucknowUniversity Studies, Faculty of Science, no. 1, 1935, reprinted in Squaring the Circle and OtherMonographs, Chelsea Publishing Co., New York, 1953.

2. E. Gouhsat, A Course in Mathematical Analysis, Vol. 1, translated by E. R. Hedrick,Ginn & Co., Boston, 1904, p. 423-425.

3. T. Bromwich, An Introduction to the Theory of Infinite Series, Macmillan & Co., Ltd.,

London, 1908, p. 490-491. Note: The proof of the sufficiency of ab > 1 + -j-(l — a) is not con-

tained in the later 1926 edition.4. G. H. Hardy, "Weierstrass's non-differentiable function," Trans. Amer. Math. Soc,

v. 17, 1916, p. 301-325.5. Nat. Bur. Standards Appl. Math.Ser. No. 5, Table of Sines and Cosines to Fifteen

Decimal Places at Hundredths of a Degree, U. S. Government Printing Office, Washington 25,D. C, 1949.

* Another counter-example found after that of the referee is the following: f{x) =

x<t>(x), x 7* 0, /(0) = 0, where <t>(x) = 1 except in the intervals {(1/n — 1/n3), 1/n], within which

<t>(x) = 0. Now f(x) is continuous at x = 0 and has no derivative there. But l/h ¡i¡t>(x)dx —► 1 as

A—»0, because the "dipped-out" area becomes an infinitesimal fraction of the whole (also in-

finitesimal) area between 0 and h, since as h ~ 1/n, we remove 22™-» l/«i' ~ l/2n* ~ 0(A).


126 HERBERT E. SALZER AND NORMAN LEVIN'E

Table of W(x) m £ cos (7nxx)/2"n-l

H'W

.000

.001

.002

.003

.004

.005

.006

.007

.008

.009

.010

.011

.012

.013

.014

.015

.016

.017

.018

.019

.020

.021

.022

.023

.024

.025

.026

.027

.028

.029

.030

.031

.032

.033

.034

.035

.036

.037

.038

.039

.040

.041

.042

.043

.044

.045

.046

.047

.048

.049

1.00000.80391.61188.53777.64747.87163.76687.60807.43502.40541.56641.54275.50694.34801.22473.27196.25665.34500.29744.19896.16232.07772.20584.28363.31741.28730.11054.17279.29881.48372.54441.32388.26990.33225.58370.74931.56632.43361.36496.55435.75565.66141.54244.36553.41175.54502.52178.49248.30088.22797

2375

00000 005S29S 4960438 5860375 2748039 38698537195734552 6194075 7806494 7631472 9336720 2791215 9825245 8791530 3945026 6887904 1848434 7809740 2002842 2654753 0175335 9744795 3456796 3647365 6016038 9729341 4294307 9259987 3384610 5821945 0999122 7813283 8474462 0492580 2930151 9109611 8586486 5826383 5098106 4042269 3761182 7067604 8053065 0690597 7446829 9660105 0388676 0264437 5019914 12

-»'(*)

1.000.999.998.997.996.995.994.993.992.991.990.989.988.987.986.985.984.983.982.981.980.979.978.977.976.975.974.973.972.971.970.969.968.967.966.965.964.963.962.961.960.959.958.957.956.955.954.953.952.951

ir M

.050

.051

.052

.053

.054

.055

.056

.057

.058

.059

.060

.061

.062

.063

.064

.065

.066

.067

.068

.069

.070

.071

.072

.073

.074

.075

.076

.077

.078

.079

.080

.081

.082

.083

.084

.085

.086

.087

.088

.089

.090

.091

.092

.093

.094

.095

.096

.097

.098

.099

.23088

.20682

.27128

.16118

.08069

.02066

.11450

.02295

.01951

.01151

.09698

.27187

.23653

.16965

.01498

.00239

.20181

.25856

.21932

.05550

.15690

.01436

.09812

.15074

.09240

.24890

.20632

.11462

.02304

.09557

.19794

.22531

.20876

.05397

.04851

.02128

.03684

.08433

.01214

.05215

.19576

.26338

.21893

.22931

.19289

.36048

.51624

.54350

.50350

.33848

91433 5352628 3971570 3134941 7156769 7012990 0455193 7365257 1972464 5568818 8014952 8035472 7200063 4563244 2143634 8770885 8095236 7431395 2357817 0433815 8095326 4783992 2003304 8856668 5088499 3558340 0759257 0033882 3518919 6081653 4901773 1998834 2059176 9457757 4366043 6319742 7858507 1837682 4933731 4221171 0122844 4174226 8160178 3650089 5854543 3494459 7668304 4209214 8710354 4006786 76

.950

.949

.948

.947

.946

.945

.944

.943

.942

.941

.940

.939

.938

.937

.936

.935

.934

.933

.932

.931

.930

.929

.928

.927

.926

.925

.924

.923

.922

.921

.920

.919

.918

.917

.916

.915

.914

.913

.912

.911

.910

.909

.908

.907

.906

.905

.904

.903

.902

.901

-W{x)


TABLE OF A WEIERSTRASS CONTINUOUS NOX-DIFFERENTIABLE FUNCTION 127

Table of W(x)—Continued

»'(») W(x)

.100

.101

.102

.103

.104

.105

.106

.107

.108

.109

.110

.111

.112

.113

.114

.115

.116

.117

.118

.119

.120

.121

.122

.123

.124

.125

.126

.127

.128

.129

.130

.131

.132

.133

.134

.135

.136

.137

.138

.139

.140

.141

.142

.143

.144

.145

.146

.147

.148

.149

.42532

.60122

.71436

.69032

.43794

.40215

.50671

.65237

.68741

.44815

.33948

.32696

.42541

.50701

.36527

.28717

.19435

.20343

.27648

.24091

.27427

.19594

.14745

.16077

.16313

.30795

.32779

.30642

.25457

.20521

.38226

.51008

.58897

.51605

.37228

.48222

.64476

.82656

.78900

.58460

.58017

.67358

.88334

.90195

.71735

.63542

.60172

.73979

.77996

.67821

1815

54041 7607728 9904664 2326595 6973064 5363534 9130937 4817461 6746475 5112393 0990492 2883696 91627682502604053 4602983 1486539 7201549 8995287 9687061 6189580 6653705 6862719 1990760 0211716 5798441 7007918 3117906 9854992 7128645 40570066053588293 0703218 2202555 4940135 7690942 4460891 9050242 1320582 1861018 6593294 6597740 7854475 2567984 3171888 1584929 4705811 3261358 5323623 64

1812

.900

.899

.898

.897

.896

.895

.894

.893

.892

.891

.890

.889

.888

.887

.886

.885

.884

.883

.882

.881

.880

.879

.878

.877

.876

.875

.874

.873

.872

.871

.870

.869

.868

.867

.866

.865

.864

.863

.862

.861

.860

.859

.858

.857

.856

.855

.854

.853

.852

.851

-ii'd)

.150

.151

.152

.153

.154

. 155

.156

.157

.158

.159

.160

.161

.162

.163

.164

.165

.166

.167

.168

.169

.170

.171

.172

.173

.174

.175

.176

.177

.178

.179

.180

.181

.182

.183

.184

.185

.186

.187

.188,189.190.191.192. 193194195

,196197198199

.60928

.48052

.49741

.49479

.47354

.49291

.36979

.30677

.21917

.21531

.32774

.30751

.27020

.11494

.05693

.19542

.30152

.38366

.22909

.09500

.17303

.33806

.55064

.46584

.30129

.27803

.39280

.65182

.65182

.53351

.44268

.43578

.61985

.64922

.62277

.54441

.43416

.47153

.44187

.48324

.48100

.36210

.28385

.13957

.17132

.24380

.21870

.13838

.09801

.14666

87419 7465499 6697079 3675145 6523522 6596963 6828855 9109741 3549907 1669983 4187639 1175250 6256659 2778174 8476441 4929668 6083944 4139179 3139440 7391416 9580098 9967973 8602832 1678110 9983741 0769565 6065936 5079852 2639388 2002371442740102187140 1521692 6387433 01654315110143722 9785912 5234653 2145544 7624639 7607250 5834378 9673511 0991207 9772532 5271322 9982360 7152327 61

897283

7385

.850

.849

.848

.847

.846

.845

.844

.843

.842

.841

.840

.839

.838

.837

.836

.835

.834

.833

.832

.831

.830

.829

.828

.827

.826

.825

.824

.823

.822

.821

.820

.819

.818

.817

.816

.815

.814

.813

.812

.811

.810

.809

.808

.807

.806

.805

.804

.803

.802

.801

-H'M


128 HERBERT E. SALZER AND NORMAN LEVIXE


ir (i)

.200

.201

.202

.203

.204

.205

.206

.207

.208

.209

.210

.211

.212

.213

.214

.215

.216

.217

.218

.219

.220

.221222

.223'224

.225

.226

.227

.228

. 229

.230

.231

.232

.233

.234

.235

.236

.237

.238

.239

.240

.241

.242

.243

.244

.245

.246

.247

.248

.249

.06366

.04175

.0747616401

.29971

.28236

.10035

.08006

.06602

.22194

.29866

.14347

.11156

.08927

.00503

.1239307237

.1287017190

.20539

.10396

.05670

.110581105322263204331389905338

.07179

.0189607113,08351.03967.28470.30373.2533112247

.1609143416

. .'.7254i ¡0020

.3929329091

. 47723

.65452

.78049

.58771

.39392

.43534

.53704

100182236470750552203581584375126743650453228839099376(5001022700561047436831819602520668816311946224935518490004979479209854220524147735227491975628982971149064204182344506421569460787746178316980375774057378802109186339027029732613946084215589270311

750418046100 !93 !29 I00 |7943079733934468328539295398414480883044757496753448858436352293482028151918968088

.800

.799

.798

.797

. 796

.795

.794

.793

.792

.791

.790

.789

.788

.787

.786

.785

.784

.783

.782

.781

.780

.779

.778

.777

. 776

.775

.774

.773

.772

.771

.770

. 769

.768

.767

.766

.765

.764

.763

.762

.761

.760

.759

.758

. 757

. 756

. 755

.754

.753

.752

.751

il' w

--dix)

.250

.251

.252

.253

.254

.255

.256

.257

.258

.259

.260

.261

.262

. 263

.264

.265

.266

.267

.268

.269

.270

.271

.272

.273

.274

.275

.276

.277

.278

.279

.280

.281

.282

.283

.284

.285

.286

.287

.288

.289

.290

.291

.292

.293

.294

.295

.290

.297

.298

.299

.70710

.59986

.42999

.36660

.32795

.45218

.43000

.38270

.32431

.18040

.20082

.19502

.28277

.33124

.19535

.13130

.05923

.20320

.38097

.36488

.30069

.12888

.21930

.45450

.58788

.61063

.37897

.35491

.53317

.74080

.87388

.66344

.55360

.59858

.75311

.93575

.80593

.70250

.62769

.64051

.76998

.71343

.70111

.59700

.48196

.48841

.43820

.53246

.50028

.36415

67811 87 I16383 91 i19525 7193235 9462544 0611134 9178607 2470934 21 !

23791 08 I34507 72 !17490 1592282 8611533 6294143 1083704 39 I01934 73 !97001 64 I81841 97 i30175 89 j72388 22 I58598 63 i96220 58 !88571 5266523 44 j37027 2878772 02 !11709 3200906 3701360 59 j38401 87 i44641 26 I41290 97 i44310 8896747 52 :92971 1968089 02 ;31523 57 i54785 6378735 1231524 3470795 5652538 50 i02427 36 I86384 17 !39792 70 !28610 21 !32711 49 !89138 70 ;47645 04 I33185 00 i

-»•(*)


TABLE OF A WEIERSTRASS CONTINUOUS NON-DIFí ERENTIABLE FUNCTION 129


.300

.301

.302

.303

.304

.305

.306

.307

.308

.309

.310

.311

.312

.313

.314

.315

.316

.317

.318

.319

.320

.321

.322

.323

.324

.325

.326

.327

.328

.329

.330

.331

.332

.333

.334

.335

.336

.337

.338

.339

.340

.341

.342

.343

.344

.345

.346

.347

.348

.349

K'(l) I

.26286

.14582

.28563

.36633

.33282

.21458

.00939

.10710

.25624

.37&38

.34385

.09875

.10737

.23i60

.45535

.54102

.33736

.28355

.30968

.51427

.66458

.55383

.51306

.43864

.52349

.63004

.59308

.62794

.51805

.47323

.45296

.41365

.52433

.45999

.37014

.22552

.10904

.23421

.25303

.23376

.05089

.15716

.09044

.01752

.09699

.01688

.25654

.27382

.21463

.00184

5556098589974075015781740963152092680078917049794520085620754648340973620026547928357867065218937371801660222208367141862442529627729659761341271473507693295766353627192012343022355545937763234293293790862906697294994215751799367209168885978285025835

6087018747304580222752197059529370205170073600981266010703890299362631195208375154729989512379645382

-ll'(x)

.700

.699

.698

.697

.696

.695

.694

.693

.692

.691

.690

.689

.688

.687

.686

.685

.684

.683

.682

.681

.680

.679

.678

.677

.676

.675

.674

.673

.672

.671

.670

.669

.668

.667

.666

.665

.664

.663

.662

.661

.660

.659

.658

.657

.656

.655

.654

.653

.652

.651

.350

.351

.352

.353

.354

.355

. 356

.357

.358

.359

.360

.361

.362

.363

.364

.365

.366

.367

.368

.369

.370

.371

.372

.373

.374

.375

.376

.377

.378

.379

.380

.381

.382

.383

.384

.385

.386

.387

.388

.389

.390

.391

.392

.393

.394

.395

.396

.397

.398

.399

H'(x)

.00778

. 17204

.23029

.23647

.01762

.06669

.00075

.02063

.08880

.03980

.10006

.12219

.18737

.10954

.12819

.07610

.09741

.22835

.19796

.16937

.00562

.07189

.04604

.07904

.09708

.12756

.30043

.27256

.21256

.09426

.26632

.48045

.55645

.52637

.33212

.39479

.54657

.66192

.69040

.49978

.48100

.50444

.56183

.62318

.,50778

.49647

.41302

.35589

.38867

.35791

77869 6722086 5021241 8630864 4425274 4222195 1362690 2175236 1213434 6477688 4060841 6989553 3455558 8321317 0087330 7038829 8956111 3766376 4135878 2955518 3033217 5357360 0108396 6658408 7084764 2811441 2250117 7267762 1078517 9894456 3963801 7819086 9323195 6152089 9951324 2974417 8707923 3384709 1653292 0537896 3238625 9379258 9005832 6858803 8026196 5333948 3252891 4092135 8382210 9263481 26

-ii w

.650

.649

.648

.647

.646

.645

.644

.643

.642

.641

.640

.639

.638

.637

.636

.635

.634

.633

.632

.631

.630

.629

.628

.627

.626

.625

.624

.623

.622

.621

.620

.619

.618

.617

.616

.615

.614

.613

.612

.611

.610

.609

.608

.607

.606

.605

.604

.603

.602

.601


130 HERBERT E. SALZER AND NORMAN LEVINE

Table of W(x)—Concluded

»'(*)

.43633

.34108

.19995

.15624

.15344

.33660

.33007

.20452

.09102

.04111

.26774

.38274

.36998

.24689

.11736

.29765

.47494

.59655

.53275

.35968

.44265

.57367

.75568

.77343

.63242

.64210

.64792

.77107

.82369

.76886

.78295

.67178

.65728

.65957

.67892

.76752

.62898

.49719

.38838

.41153

.58326

.52360

.38190

.17443

.12778

.32413

.38999

.35689

.13266

.00059

89981 2564853 6766617 1484342 3931864 8942737 8906597 3746632 0642265 6867965 8344625 3333375 0951442 4245307 0078392 8417844 1121492 2284918 7436804 5931001 5428777 2468152 0010645 5316806 9361726 8294688 1512620 6483101 9179240 7158123 18,98538 2995122 9968184 1056572 4226214 6041703 5665506 2039800 0606903 2659153 1416692 2470974 048298601231794416866244407 0824535 2784382 2006984 60

12017158

-H'(x)

.600

.599

.598

.597

.596

.595

.594

.593

.592

.591

.590

.589

.588

.587

.586

.585.584.583.582.581.580.579.578.577.576.575.574.573.572.571.570.569.568.567.566.565.564.563.562.561.560.559.558.557.556.555.554.553.552.551

.450

.451

.452

.453

.454

.455

.456

.457

.458

.459

.460

.461

.462

.463

.464

.465

.466

.467

.468

.469

.470

.471

.472

.473

.474

.475

.476

.477

.478

.479

.480

.481

.482

.483

.484

.485

.486

.487

.488

.489

.490

.491

.492

.493

.494

.495

.496

.497

.498

.499

.500

H'(x)

.14085

.28701

.40662

.26053

.09792

.14569

.26984

.47876

.44907

.34753

.33327

.35146

.52273

.55987

.57344

.57404

.48136

.51427

.51301

.61436

.69144

.56392

.45520

.32911

.42540

.57627

.51397

.35913

.10430

.10454

.26292

.31706

.24133

.05762

.17288

.08627

.05153

.12044

.10705

.26694

.28240

.15028

.02361

.06684

.15875

.23215

.18367

.01931

.00378

.04441

.00000

88911 0785703 9430552 6058919 4469986 3354090 9709362 8524067 9728535 1636936 0019438 2325764 1162074 4341267 6706332 4569550 4222660 4261901 2735662 5117729 6970666 0563767 9204387 3592328 1032355 3907637 4183689 3637922 9850805 9999179 6626878 6781976 5462343 5350734 01120307988393039 5948896 5903214 6606923 2683062 1616426 9575574 0238933 2942472 1263219 7546210 9221600 0755928 2166347 1100000 00

1517

.550

.549

.548

.547

.546

.545

.544

.543

.542

.541

.540

.539

.538

.537

.536

.535

.534

.533

.532

.531

.530

.529

.528

.527

.526

.525

.524

.523

.522

.521

.520

.519

.518

.517

.516

.515

.514

.513

.512

.511

.510

.509

.508

.507

.506

.505

.504

.503

.502

.501

.500

-H'W


Table of a Weierstrass Continuous Non-Differentia ble Function

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Transcript of Table of a Weierstrass Continuous Non-Differentia ble Function