Electrochemical generation and recapture currents are linked by Hilbert transformation

15

J Eleetrcunal Chem, 218 (1987) 15-28 Filsewer Sequoia S A, Lausanne - Pnnted m The Netherlands

ELECI’ROCHEMICAL GENERATION AND RECAPTURE LINKED BY HILBERT TRANSFORMATION

CHRISTA

L COLYER,

MICHAEL. R HEMPSTEAD

l

CURRENTS

ARE

and KEITH B OLDHAM

Trent Unrversrly, Peterborough (Canadn) (Recewed 22nd July 1986)

ABSTRACT We have demonstrated expenmentally that It IS possible to construct recapture currents accurately from generation currents Obstacles Impede the more useful transformation of recapture currents mto generatlon currents

INTRODUCTION

In this article we are concerned Hrlth a soluble species Y that 1s lmtlally absent from a solution that contams a precursor species X. The electrochenucal reactlon X(soln)

+ ng e- --* Y(soln)

(1)

that generates Y from X may be ather a reduction or an oxidation, ns bemg negative m the latter case Reaction (l), commencmg at t = 0, 1s not necessanly reversible, nor do we specify the type of electrochenncal expenment that leads to tbs reaction. Thus the form of the generation current 19(t) 1s not prescribed Durmg this generation process, species Y Qffuses away from the electrode and becomes dispersed m the surroundmg solution Species Y 1s itself electroactlve by virtue of the reactlon Y(soln)

+ nr e- + Z(soln)

where, agam, n, may be posltlve concludes Hrlth a phase, t > T, m extreme that reactlon (2) proceeds polanzatlon The current l,(f) that

l

(2) or negative We specify that our expenment whch the electrode potentlal 1s sufflclently under conditions of complete concentration flows durmg this recapture phase results from

Present address Department of Chennstry, York Umverslty, Downswew, Canada

0022-0728/87/$03

50

0 1987 Elsewer !3equola S A

16

the back dlffuslon of Y towards the electrode because the surface concentration of Y has been made zero. The simplest example of a generation/recapture sequence occurs when reactlon (2) 1s the reverse of reactlon (1) Then X and Z are Identical and ns + nr c- C More generally, however, ns and nr may have slrmlar sign or opposite signs but unequal magmtudes In this article we demonstrate that a general relation&p exists between l,(t) and l,(t), and that this relation&p mvolves Hdbert transformation. We show how rJt) may be denved if r,(t) IS known. The Inverse, but more useful, procedure of determmmg z,(t) from z,(r) 1s shown to be more dlfflcult A future article wdl address methods by which the mverslon may be attempted.

INTERRELATIONSHIP

OF THE CURRENTS

If the generation of Y begms at tune r = 0 and proceeds untd time t = T, then the recapture current 1s related to the generatlon current by the deflmte mtegral

G(t) =

nr sn,(t

- T)l’*

T

/0

J,(T)@--

$”

d7

t--7

wbch 1s apphcable for any tune c greater than T The only restnctlons embodled w&m this expression are that (1) both the generation and recapture reactions must be electrochermcal, (u) species Y must lmtlally be absent from solution, (m) the recapture reactlon must occur under conditions of complete concentration polmtlon, and (IV) transport must be by semunfmte planar dlffuslon The proof of thy expresslon 1s provided m Appends A by a rather lengthy exercise m Laplace transformation. Although the development of the mterrelationsbp between the generatlon and recapture currents 1s rather mvolved, the fmal expresslon 1s versatile Expression (3) enables the recapture current to be calculated by knowmg only (1) the generatlon current as a function of time, (u) the duration of the generation phase, and (m) the ratio n,/n,. Restictlons have not been placed on the signal that 1s imposed durmg the generation phase or on the electrochemical reverslbhty of the generatlon or recapture reactions Nor have Qffuslon coefflclents of &ssumlar species been assumed equal. Thus eqn. (3) is more general than many relationshtps encountered m electrochermstry For most generation/recapture experrments, the generation and recapture reactions are mutual converses The ratio n,/n, IS mmus one for these reaction schemes

17

and hence eqn (3) may be rewntten as

J,(t)

=

T l,(7)@-

-1 r(l-

T)“’

/0

W2

d7

(4)

f-r

Thus expresnon IS proven easdy m Append= B wth the sud of the fractlonal calculus Although this second proof was not necessary to obtam eqn. (4), It dlustrates that the fractlonal calculus can often be advantageous

GENERATION/RECAPTURE

EXAMPLES

Support may be gamed for the relation&p gven as eqn (4) by consldenng documented expenments which follow a generation/recapture reactlon scheme Two examples wdl suffice The recapture current for each wdl be formulated by using eqn (4) and compared urlth hterature expresslons. Cottrelhan generatlon / recapture The current response dunng the generatlon phase of a Cottrell expenment gven by

IS

PI

zg(t) = nAFc( D/w)“’

(5)

where the symbols have then usual slgmflcance eqn. (4) yields

G(t) = _ EfE(

Substitutmg thus expresmon mto

2_)“‘/,‘( ~y2-k_

(6)

The defmte mtegral m thus expression may be evaluated by the substltutlon sm28 = r/T which leads to the result [2] G(t) =

1 (t _ T)li2

--

1 t’12

1

(7)

which was denved by Kambara [3] for so-called “double step voltammetry” Galvanostatrc generatlon /recapture Consider an expenment m wluch species Y IS generated at an electrode by a current of magmtude I that remams constant durmg the mterval 0 < t -C T Thu may reqmre that T not exceed the transItion time of the system [4] The recapture current may be expressed, from eqn. (4), as r,(t)

= -

i/,‘( c2)1’2&

18

which may be evaluated by the substltutlon z* = (T - ~)/(f - T) to yield

which 1s consistent Hnth the solution obtamed from the theory of the electrolysis of non-umform solutions [5] If errors found m that article are corrected CONSTRUCTION

OF RECAPTURE

CURRENT

FROM GENERATION

CURRENT

A generation/recapture expenment may be dlvlded mto three time periods Imtlally, that 1s for t c 0, the system 1s at equlhbnum durmg which nothmg happens Then, durmg 0 < t < T, species Y 1s created m the generation phase Finally, for t > T, species Y 1s consumed m the recapture phase The symmetry of the problem may be enhanced by conmdenng the nudpomt of the generation phase, that 1s t = T/2, as the time ongm The new ongm 1s estabhshed by the deflmtions g=r=(2t/T)-1

(10)

of two new undlmenslonahzed mdependent vanables g and r. Although g and r are Identical, it wdl be convement to use g as the Independent vmable of the generatlon process and r as the mdependent vanable of the recapture process Equation (3) may be expressed m terms of the new independent vmables as 1

“r

Jr(r) = ng7r(r-

l)l’*

J-1

(1-

S)"*J,(d r-g

01)

dg

m which r must be greater than one to descnbe the current m the recapture phase The mtegral m eqn. (11) closely resembles a Hdbert transform [6] The Hllbert transform has rarely been apphed m the analysis of physical systems However, there have been some noteworthy apphcations [7-91 of which the automatic phase correction of magnetic resonance spectra [lo] 1s probably of most mterest to chenusts The Hllbert transform of a function f(x) of the vmable x, 1s a function of a second vmable y and IS defined by F(y)

= H{f(x)}

= ;f_“,$+

02)

where the integral 1s replaced by Its Cauchy Prmclpal Value [ll] to avold the mfmte dlscontmmty at y = x Thus, it becomes convement to define a “generatlon functlon” by the tnparhte deflmtlon g<

0

G(g) = (1 - g)“*J,(g)/n, 0

-l-=g<

-1 1

g’

1

(13)

19

whereby we see that (r - 1)1’2J,(+J,

r>l

= H{‘h))

04)

Hence, d we call the left-hand side of eqn. (14) “ the recapture function”, we see that It 1s simply the Hdbert transform of the generatlon function The lmplementatlon of eqn. (11) reqmres the construction of an algonthm whereby values of I, may be found from measurements of Jo Nowadays electrochermcal data are usually collected dIptally at evenly spaced times Letting A be the time between current measurements and I, represent the current measured at time I,, eqn (11) may be approxnnated by Jr(t)

=

nr ~(a,(~)+s)[(T-t,)‘/*-(T-t,+,)‘/* n,lzA( t - T)“*,-O -(t

- T)‘+mn(

s)l’*

- arctan( :--‘;-“)“*}]

-~[(T-r,)3/*-(T-r,+~)3’2] where a, = I, + 1 -I, and /3, = J,(2t,+l - T)/TJ,+1(2t, - T)/T wth to=& t,+l = T, a,, = a1 and & = &. The construction of this algonthm 1s described m AppenQx C and 1s based on a plecewse-hnear approxlmatlon of the generation current. Experimental Chronoamperometnc Fe(CN)iFe(C,O,):-

expenments were performed on the systems

+ e- + Fe(CN)d+ e- + Fe(C,O.,)i-

(16) (17)

at an mliud cvcular platinum electrode (Pme Instrument Model DT6) of area 46 1 mm* usmg solutions described m the legends of Figs. 1 and 2 Cychc voltammograms were obtamed for each system to confirm that eqn (16) occurs reversibly whereas eqn (17) does not Each system was then stu&ed using seven &fferent potential signals, namely double step, ramp step, cychc ramp (1 1, 1 5 and 1 10 symmetrres), ramp-hold-step and multistep The expenments were performed using a Prmceton Apphed Research Corporation Model 170 electrochenustry system hnked to a Hewlett-Packard Model 3497A data acqulsltlon system controlled by a Hewlett-Packard Model 85A personal computer Currents were recorded at 20,160, ms durmg both the generatlon and recapture phases. The generatlon 300,440, data were used m conJunctlon ~th eqn (15) to compute recapture currents Two typical examples of such compmsons are shown as Figs. 1 and 2 The excellent agreement evldent m these two diagrams confirms the vahQty of the Hllbert transform method of calculating recapture currents from generatlon

20

400

a

* + *

* **

200

-

’

**** ****-A

\

I

0

:

-200. <

‘9

-4000

5

t/s

10

15

Fig 1 The asterisks represent current versus time data from a double potentml step expenment performed on a deaerated 5 0 m M solution of K,Fe(CN), prepared m a supportmg electrolyte of 2 0 M KCI The electrode potential was changed from +0 65 V (versus a saturated Ag/AgCl/KCl electrode) to -015 Vat time r-0 and from -015 V to +065 V at time t=84 s The full hne was obtamed by IWbert transformation as described m the mam text

*** * * * * +

l9 *

*

+ *

*

*

*

*

*

1,

* *

Fig 2 The aster&s represent current versus hme data from a cyclovoltammetnc expenment performed on a deaerated 5 0 mM solution of Fe(C,O,):prepared m a supportmg electrolyte of 10 M K&O, and 0 10 M H&O, Startmg at +0 15 V (versus a saturated Ag/AgCl/KCl electrode) at time t = 0, the electrode potential was ramped at a rate of - 98 6 mV/s to - 0 45 V and was then reversed at time z = 6 1 s until the potenhal returned to +0 15 V after which It was held constant The full lme was obtamed by I-hlbert transformation of the generation current whch was consldered to end at time t-1217s

21

currents and demonstrates that the procedure 1s mdependent of the reversibihty of the system and the nature of the signal. RECONSTRUCTION

OF GENERATION CURRENT FROM RECAPTURE CURRENT

The abihty to construct the recapture current from the generation current offers few practical apphcauons, whereas the inverse of this procedure has great potential. Constder the electroanalysts of substance X by some voltammetnc procedure that generates Y The presence of an mterfenng substance (such as oxygen m the case of an electroreduction) will often obscure the voltammogram of X If, however, Y is electroactrve m a potential range free of interference, the abihty to construct the mterference-free voltammogram of X from the recapture current of Y has great utrhty Accordmgly, we have devoted much effort m seekmg a way of developing a method for deternnmng Jo values from I, data. At first sight, the properties of the Hrlbert transform seem ideally suited to thrs task. The so-called skew-reciprocity relationship of Hilbert transforms [12] states that if F(y)

then

= H{f(x))

f(x)

= -H{F(y)]

(18)

Having defined the left-hand side of eqn (14) as the recapture function R(r)

(1 -

d”‘J,(d

= G(g)

=

-H{R(r)}

= -H

(1 -

r)“2J,(r)

“r

54

then (19)

suggestmg that the generation current may easily be found from measured values of the recapture current I,. However, smce the Hilbert transform involves an mtegral from - cc to co, values of~,areneededmthephasesr~-1and-1~r~1aswellasmtheexpenmentally accessible regton r > 1 That is, there exist two purely fictttious functions that must be known m order to accomphsh the transformation mdtcated m eqn (19) We know of no way of fmdmg these fictitious functions m the general case If the form of the generation current is known then one may compute the fictitious functions. For example, with Cottrellian generation, g<

(0

-1

l/2

G(g)=

(20)

-ll

where k = r~AFc(2D/lrT)‘/~,

one finds the recapture function to be r< -l
-1

(21) r>l

22

Generally, however, the flctltious currents wdl remam unknown, thus Impedmg the inverse transformation We have mvestigated several strateges for clrcumventmg our ignorance of the flctltlous currents [13] but without success ACKNOWLEDGEMENT

We urlsh to thank the Natural Sciences and Engmeermg Research Council of Canada for an undergraduate award (to Chrrsta L. Colyer), a graduate award (to Michael R Hempstead) and an operatmg grant (to Keith B Oldham) APPENDIX

A

A proof of eqn (3) IS presented here which rehes upon the operation of Laplace transformation Let c(x, t) denote the concentration of species Y at a &stance x from the electrode at a time t m the interval 0 -c t -c T The followmg condltlons can be apphed to Y durmg the generation phase: (I) Y IS transported m accordance wth Flck’s second law of diffusion (A 1) (II) ~mtmlly, Y IS absent from solution c(x, 0) = 0 (m) at a suffxxent solution

distance from the electrode,

(A-2) Y wdl remam absent from

c(o0, t) = 0

(A 3) (IV) the total number of electrons transferred at the electrode m a umt tune must be propotional to the quantity of Y that reaches the electrode m that time penod

$0,

t) = - ~

go

(A-4)

n,AFD

The Laplace transform of c(x, t) Hrlth respect to t urlll be denoted E(x, s), s bemg a dummy vanable Transformation of eqn (A 1) followed by substitution of eqn (A-2) yields D-$E(x,

s)=sZ(x, s)-c(x,O)=sC(x,

Transformation

(A 5)

of eqns (A.3) and (A.4) gves

?(co, S) =o

-&c(o, s) =

s)

(A 6) - *

J-(4 s

where l,(s) IS the Laplace transform of l,(t)

(A 7)

23

Equation (A 5) 1s a second order ordmary dfferentlal equation that can be solved m terms of two arbitrary constants B,(s) and B,(S) as E(x,

3) = B,(s)

exp 1 -(s/D)“*x]

+ B,(s)

but, d eqn (A 6) 1s to be satisfied, B,(s)

&E(x,s) = -(s/D)"*Bl(s)

exp[(s/D)“*x]

(A 8)

must be zero Dlfferentlatlon then gves

exp[ -(s/D)“*x]

(A 9)

whch upon comparison urlth eqn (A 7) establishes B,(s) Equation (A 8) may now be wntten as

as I~(s)/[~~AF(sD)‘/*]

exp[ -(s/D)“*x]

(A 10)

(D.s)l’*

Inversion of eqn (A-10) 1s possible wth the iud of the convolution mtegral of Laplace transformation [14]. Apphcatlon of this integral to eqn (A 10) gves

[&(t

\

7)]1,2exp (4D;:lr)))

d7

=

(A.ll)

The concentration profile durmg the generatlon phase cannot be determmed from eqn (A 11) unless the functional form of r,(t) 1s known. However, ths equation 1s useful m Its present form smce It 1s applicable to any generation current Now consider the recapture phase of the expement Condltlons (A 1) and (A 3) are still apphcable but (A 2) and (A 4) are not, and therefore must be replaced The recapture reaction 1s lmtlated at time t = T. At this Instant, the concentration profile 1s gven by c(x,

1

T)= n,AF(

T

‘g(7) / 0 (T - #*

,D)“*

exp( ,,I,*1

TJ)

d7

(A 12)

accordmg to eqn (A 11). A new boundary condlhon, c(0,

t) = 0

(A 13)

now arises as a consequence of Y bemg consumed under conditions of complete concentration polanzation for t > T The recapture current 1s gtven by z,(t) = n,AFD$c(O,

t)

(A 14)

which 1s the analogue of eqn. (A 4) m the recapture phase. Therefore, the evaluation of the recapture current does not reqmre a complete expression c(x, t) for the concentration profile, only the surface gra&ent ik(0, t)/ax need be known

24

Usmg u as the dummy vanable for Laplace transformation with respect to t - T, the transforms of eqns. (A l), (A.3) and (A 13) are D d2 _ -c(x, dx2

u) = uC(x, u) - c(x,

T)

(A 15)

E(o0, u) =o

(A 16)

and c(0, u) = 0

(A 17)

Equation (A 15) IS an mhomogeneous second order ordmary different& that can be solved by standard methods [15] to gve Ecx

,

u)

=

ev[ -WW2x] [4(a) +~xchT) exp[(~/~)“2p] dp] 2(Du)l/2

+exP[ b/W2x] [B,(u)

- iXc(pY

2( Du)“2

where B,(u) that B,(u)

and B2(u) are arbitrary

= imc(p,

,

u)

=

T) exp[ - (u/@“2p]

exP[-WW2x]

_ exp[ - ( u/D)1’2x] /,

2( Du)1’2 + exp~(~~~~xl

dp]

dp

(A 19)

as the ne@ve

x 1 C(P, 0 0

2(Du)“*

T) exp[ -(u/D)‘/2p]

Apphcatlon of con&tlon (A-16) estabhshes

while con&tlon (A.17) estabhshes B,(u) may be wntten as qx

equation

of B2(u)

exp[(u/@“2p]

dp

00 C(P, T) exp[ -(u/D)‘/‘p]

Imc( p, T) exp[ - ( u/D)“‘2p]

Hence eqn (A 18)

dp

dp

(A 20)

x

U

Dlfferentlation of eqn. (A 20) with respect to x leads to ZD-&C(x,

u) = -exP[

-(u/D)1’2x]ixc(p,

+exp [ -(c~/O)l’~ + exp [ (u/D)“’

x] Jrnc(p, x] /“(p, X

T) exp[(u/D)“‘p] T) exp[ - (u/D)“2p] T) exp[ - (u/D)“‘p]

dp dp dp

(A 21)

25

which slmphfles to -&C(O,

u) =

hlmc(x,T)

exp[ - (o/D)“*x]

dx

on settmg x equal to zero Substltutlon of c(x, &o,

u) =

1 n AF&*D3/*

xi-exp{

(A 22)

T) from eqn (A 12) gves

Q(T) / oT (T- T)l/*

-x($)“‘-

4D(z_

T)}

dx cl7

(A 23)

after the order of the two mtegratlons 1s reversed The mner mtegral may be more easily evaluated after a change of vanable to { a(T - ~)}l’* + x/[2{ D(TT)}‘/*] Thus leads to

&E(O, u) = &lTl,(*)

exp[u(T-T)]

erfc[u(T-7)]1’2

d7

s

as the final expression for the transform of the surface concentration durmg the recapture phase Tables of Laplace transforms [16] gve the mveruon L-‘{exp(ks)

erfc(ks)“*}

= ,(,:

k) (5)“’

(A 24) gradlent

(A 25)

for k a positive constant Recalhng that u 1s the dummy vanable correspondmg to t - T, the mverslon of eqn (A 24) IS $0,

1

t) =

T+)(T-~'*~~

an,AFD(t

- T)“*

J0

t-7

(A 26)

Combmation of ths expresslon Hnth e.qn (A 14) leads to J,(r)

=

n, sn,(t

- T)“*

~J1(7)(T--d'*~~

/0

t-7

(A.27)

whch completes the proof APPENDIX

B

A proof of eqn. (4) 1s proved here usmg the fractional calculus It wdl be assumed that the generatlon and recapture are mutual converses This generatIon/ recapture scheme may be expressed as X+ne-+Y

(B 1) where n IS posltlve for a reductive -generatIon reactlon, and negative for an oxtdatlve one

If an electroactive species, imttally present at a umform concentration c*, is transported to and/or from an electrode by semunfmte planar dlffuslon, then its concentration at the electrode surface 1s gven by [17] c(0, t) = c* + m(t)/&W””

(B 2) where m(t) IS the faradsuc senumtegral and D 1s the diffusion coefficient of the electroactive species The faradiuc semnntegral may be defined as ,-j-‘/2

dr=- dl-‘,2 I(‘)

(B 3)

where I 1s the faradzuc cathodic current flowmg at time t For species Y m this eqmhbnum/generatlon/recapture expenment, c* 1s zero In ad&tlon, the recapture reaction proceeds under condltlons of concentration polmzation and hence ~(0, t > T) 1s also zero Therefore, it follows from eqn (B 2) that m(t > T) must also be zero. The convolution integral defmmg the senumtegral m eqn (B 3) may be split mto two components thereby defimng m(t > T) as T) = ‘/’

m(t>

C2

J(r)

dT+-

0 (t - #‘2

1

J(T)

r

dT=O

a1’2 J T(t+1’2

@

4)

The second of these integrals represents the operation of semnntegratmg the current starting at time t = T and therefore it may be referred to as the “recapture semnntegral”. This integral may be wntten m the notation of the fractional calculus as d-*/2

m’(f)

= [+

1

_ T)] -1,2 ‘(‘)

=

&2

Jb)

* /T

tt

_

41,2

(B-5)

dT

It then follows that d-‘/2

1

[d(t_T)]-‘/2J’t)=

-61/2

= J(7) J o (f_T)‘/2d7

*‘T

(B 6)

by combmmg eqns. (B-4) and (B.5). In the same way that the derrvatlve of an integral of a function 1s the function itself, the sermdenvative of the semuntegral of a function wdl also be the function itself [18] Therefore, if eqn (B 6) 1s sermdfferentiated usmg T as the lower hnut, then &/2

r(t>i")=

a”’

[d(r-

T)]“’ d*/2

[d(t

27

z)-l12 may be found m standard tables [18] to be [(Kz)‘/~(~ + z)]-’ and hence the embraced term m eqn (B 7) 1s (T - T)‘/*/[{ s(i - T)}“‘( t - T)]. Equation (B 7) then becomes r(t>

T)=

1

T1(7)(T7)1’2d7 j t-7 7r(t - T)l’* 0

-

@

8)

which, apart from mmor notational differences, IS ldentlcal to eqn (4) and hence completes the proof APPENDIX

C

We assume the generation current measurements I~, 12, , I, are taken at equally spaced instants, so that tJ = t, + (J - l)A but, to match expenmental hnutations, t, IS not necessanly equal to A nor 1s t, necessanly equal to T The mtegral m eqn (11) IS first spht mto J + 1 umts, (1 - s)“21,(s) r-

-

g

10

dg+j

r-g

dg

d1’2$(d dg

g1

s)“21,(g)

-

I

r-g

Cl)

where g, = (2$/T) - 1 The implementation of eqn. (C 1) would reqmre contmuous measurement of us, whereas m fact we know the generation current only at pomts correspondmg to g,, approxlmatlon, illustrated m Fig. 3, IS therefore made m which a g,* -9 g, An

-1

gp

t

1

I\

‘3

I 0

t3

0

‘4

1

l9

‘._ ‘J-2

T/2

:

T

‘t.

FIN 3 Dmgram hstratmg the “pwcewse-hear approxlmahon” that was used by the I-hlbert transformation algorithm to convert generation current data mto the predxted recapture current

28

hnear relation&p is assumed between adjacent pomts Thus a typical umt in eqn (C.l) IS replaced by s,+l

J gJ

where

(1 -

d1'2J,(d

and /3,=g,+,

(Y,=J,,.~-J~

g,+, (1- d"*J,(d J g1

(C 2)

r-g

r-

g

dg=

T(aJ;+

J, -

g+, + 1,

‘J)

- (r - I)“*(

which on evduatmg the integrals gWes

[(I _ gJ)1’2 _ (1 - g,+1)1’2

arctan( s)“*

-~[(l-gJ)3’2-(l-gJ+,)3’2]

- arc+

’ r_“:”

,“*}I

(C 3)

Combmatlon of thus result with eqn (C 1) leads to eqn (15) of the mam text when the two atyplcal umts are treated appropnately REFERENCES 1 F G Cottrell, Z Phys Chem, 42 (1903) 385 2 I S Gradshteyn and I M Ry&k, Tables of Integrals, Senes and Products, Acadenuc Press, New York, 1965, Section 3 615, eqn 1 3 T Kambara, Bull Chem Sot Jpn, 27 (1954) 523 4 A J Bard and L R Faulkner, Electrochemxal Methods, Wdey, New York, 1980, p 253 5 K B Oldham, Anal Chem (40 (1968) 1799 6 A Erdely, W Magnus, F Oberhettmger and F G Tnconu, Tables of Integral Transforms, Vol 2, McGraw-Hdl, New York, 1954, pp 239-262 7 A Papouhs, Systems and Transforms with Apphcahons m Optics, McGraw-Hdl, New York, 1968, p 435 8 SC Mehrotra, J Quant Spectrosc R&at Transfer, 32 (1984) 169 9 J Irvmg and N Mullmeux, Mathematics m Physics and Engmeermg, Acadenuc Press, New York, 1964, p 633 10 RR Ernst, J Magn Resonance (l), 1 (1969) 7 11 M L Boas, MathematxaJ Methods m the PhysIcal sciences, 2nd ed , Whey, New York, 1983, p 606 12 A Erdblyl, W Magnus, F Oberhettmger and F G Tnconu, ref 6, p 239 13 M R Hempstead, M Sc Thesls, Trent Umverslty, 1983 14 R V Churchdl, Operational Mathematxs, McGraw-I-I& New York, 1958, pp 45-48 15 G M Murphy, Ordmary Differential Fiquatxons and Tkar Solutions, Van Nostrand, Pnnceton, 1960 16 M Abramov&z and I A Stegun (Eds ), Handbook of Mathematical Functions, National Bureau of Standards, Washmgton, DC, 1964, eqn 29 3 114 17 K B Oldham and J Spamer, J Electroanal Chem ,26 (1970) 331 18 K B Oldham and J Spamer, The Fractional Calculus, Acadenuc Press, New York, 1974