January 2017

NASA/CR–2017-219017

New Mathematical Functions for Vacuum System Analysis

Michael S. Woronowicz
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January 2017

Michael S. Woronowicz

SGT, Inc., Greenbelt, MD

NASA/CR–2017-219017

New Mathematical Functions for Vacuum System Analysis
Trade names and trademarks are used in this report for identification only. Their usage does not 

constitute an official endorsement, either expressed or implied, by the National Aeronautics and 

Space Administration.

Acknowledgments

The author gratefully acknowledges support from NASA Contract NNG15CR64C, 

Eve Wooldridge and Randy Hedgeland of NASA-GSFC, Code 546, Dong-Shiun Lin and 

Ray LeVesque of SGT, Inc., and Martin Grabe of DLR-IAS.

Level of Review: This material has been technically reviewed by technical management.

Notice for Copyrighted Information

This manuscript was written by employees of SGT, Inc., in Greenbelt, Maryland, under 

NASA Contract Number NNG15CR64C. The United States Government has a non-exclusive, 

irrevocable, worldwide license to prepare derivative works, publish, or reproduce this manuscript, 

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1

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Abstract

A new bivariate function has been found that provides solutions of integrals having 

the form u e du −η u which arise when developing predictions for the behavior of pressure 

within a rigid volume under high vacuum conditions in the presence of venting as well as 

sources characterized by power law transient decay over the range [0,1) for η and for u

> 0. A few properties of the new function are explored in this work. For instance the η = 

1⁄2 case reproduces the Dawson function. In addition, a slight variation of the solution 

technique reproduces the exponential integral for η = 1. The technique used to generate 

these functions leads to an approach for solving a more general class of nonlinear 

ordinary differential equations, with the potential for identifying other new functions that 

solve other integrals.

Introduction

The evolution of pressure versus time p(t) within a rigid volume V with venting in free 

molecule flow becomes governed by the behavior of gas sources Q contained within the 

volume. Based upon a mass conservation statement for an ideal gas at room 

temperature T [1] 

= Q − C(p − p∞ ) dt

dp V . (1)

Venting is described by conductance C to exhaust pressure p∞ . For molecular flow 

conditions, C is not a function of pressure, and for this discussion it is assumed p ∞ = 0. 

Dividing by C and defining characteristic timeτ ≡V C and reduced gas load q ≡ Q C ,

q(t) p dt

dp τ = − . (2)

Let s ≡ t τ . For sources associated with diffusion-limited outgassing, classical 

theoretical behavior yields q ∝ 1 s , [2] while for those associated with surface 

desorption q ∝1 s . [3] While these sorts of behaviors are often observed 

experimentally, it appears more generally one must deal with [4]

( ) η

s

c

q s = , (3)

where c and τ are constants for a given system and 0.5 < η < 1.0, although this 

parameter range does not represent hard limits. Rearranging and introducing integration 

factor s e
2

( ) η

s

e

pe c

ds

d s s = . (4)

The solution to this ordinary differential equation for pressure p(t) with some initial 

value p1at 1t becomes

( ) ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ − ∫ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = + ∫ −Δ − −Δ − 1

1

0 0

1

s u s s s u s s du

u

e du e e

u

e

p t p e c e

η η . (5)

The matter now becomes one of evaluating the parenthetical expressions.

Analytical Development

It appears no general solutions for integrals of the type highlighted in Eq. (5) have 

ever been published, as they are conspicuously absent from references on mathematical 

functions. When η is a positive integer their behavior is governed by the exponential 

integral Ei(s), [5-7] and the Dawson function D( s ) describes η = 1⁄2. [7-9] For this 

integral there also appears to be a relationship with the lower incomplete gamma 

function under certain conditions. [10-12]

The procedure for direct numerical solution of this integral seems to be quite 

problematic. The integrand has a singularity at s = 0, and for large arguments both the 

numerator and denominator increase without bounds. On the other hand the scaling 

factor ahead of the integral counteracts some of this behavior overall. 

When 0 < η < 1, let b x ≡ u , where b ≡ 1 (1−η). Then

du be e dx bG( x)

u

e

e x x x s u s b b, 0 0η η ∫ = ∫ ≡ − − . (6)

It happens that G(η, x) satisfies its own ordinary differential equation:

1 1 + = ∂

∂ − bx G

x

G b . (7)

Properties of Bivariate Function G(η, x)

The solution of Eq. (6) may be accomplished much more easily numerically using 

Eq. (7) than it can by attacking the integral directly. Eq. (7) may be explored to discover 

a variety of properties of G.
3

Integration of Eq. (7) for G across various values of η and x produces curves like 

those presented below in Figure 1.

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10

G(η,	x)

x

G(0,	x) =	1-e-x

G(0.5,	x) = D(x)

G(0.1,	x)

G(0.3,	x)

G(0.9,	x) G(0.7,	x)

G(0.9999,	x)

FIGURE 1. Function G(η, x) for selected values of η.

When (η, b, x) = (0, 1, s), G(0, x) reproduces expected limiting behavior of

( ) s s s s s s e e ds e e e − − − ∫ = − 1 = 1− 0

, (8)

and when η = 0.5 the function is equivalent to the Dawson function D(x), with its 

maximum value of about 0.54 at x ≈ 0.92. [7-9]

As η →1, b → ∞ , and the function steepens to produce a sawtooth shape with 

infinite slope at x = 1, which appears to anticipate the discontinuity associated with the 

exponential integral. [2]

Also apparent in Figure 1 is that Gʹ (η, 0) = 1 at x = 0, independent of η. This 

property is also observed by setting x = 0 in Eq. (7). In terms of physical units, this 

property indicates that regardless of η and the fact that the diffusion-desorption 
4

mechanism starts with a singularity at t = 0, pressure within the volume is initially wellbehaved with no such influence.

Less obvious in Figure 1 is the fact that for large values of x independent of η:

( ) 1

1 , − → ∞ → b bx

G η x . (9)

In terms of satisfying Eq. (6), recall the solution for the scaled integral was defined by 

bG with b s = x . Substitution of Eq. (9) into Eq. (5) for large s = t τ leads to

( ) q(s)

s

c

p t → ∞ → = η . (10)

Eq. (10) is the solution to Eq. (2) when p! = 0.

0

0.25

0.5

0.75

0.9999

η

x

0-0.1 0.1-0.2 0.2-0.3 0.3-0.4 0.4-0.5 0.5-0.6 0.6-0.7 0.7-0.8 0.8-0.9 0.9-1

G	(η,	x)

FIGURE 2. Contour map of function G(η, x).

A contour map for G(η, x) is presented in Figure 2 essentially spanning the range for 

η and computed for x out to three in increments of (Δη, Δx) = (0.01, 0.01). The linear 

slope in the neighborhood of x = 0 appears as a series of equally-spaced vertical bars, 

and the discontinuity associated with the exponential integral is present as a node at (η, 

x) = (1, 1). The prominent freeform orange cross around x = 0.8 is characteristic of a 

saddle point. Owing to the simplistic integration associated with this figure, one may say 

the saddle point occurs approximately at (η, x) = (0.56, 0.85) with G ≈ 0.54.

A three-dimensional representation of Figure 2 is depicted below in Figure 3 with the 

η scale reversed to better descry the approach to the singularity at (η, x) = (1, 1). 

Figures 1 & 3 also reveal that another singularity is approached as (η, x) → (0, ∞ ), 

where G(0, ∞ ) → 1, but G(0+, ∞ ) → 0.
5

0

0.25

0.5

0.75

0.9999

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

G (η,	x)

x

0.9-1

0.8-0.9

0.7-0.8

0.6-0.7

0.5-0.6

0.4-0.5

0.3-0.4

0.2-0.3

0.1-0.2

0-0.1

FIGURE 3. Three-dimensional contour map of function G(η, x). (η-axis reversed from 

Fig. 2.)

Scaled Exponential Integral Solution F(y) Properties

Clearly the change of variable introduced to produce Eq. (6) will not work for η = 1, a 

case corresponding to surface desorption. [3] Instead let y ≡ lns . Then

du e e dy F(y )

u

e

e

y e e

s u s y y

∫ = ∫ ≡ −∞

− −

0

. (11)

Eq. (11) would generally be described by an exponential integral Ei(ξ), specifically

( ) ( ) e y y e e F y e e dy e e

y y y

≡ ∫ = − − −

−∞

− Ei . (12)

In fashion similar to Eq. (7) it is determined that

+ e F = 1

dy

dF y . (13)

From Eq. (13) it can be seen that F'(y → −∞)→1, which corresponds to s → 0 . It 

resembles behavior observed for G(η, 0). Also, for large y, analogous to Eq. (9), 
6

( ) s

F y e y 1 → ∞ → = − , (14)

and in physical units

( ) q(s) s

c

p t → ∞ → = . (15)

Function F(y) is depicted in Figure 4. Its maximum value is approximately 9.60 for 

y ≈ −2.27, ( s ≈ 0.103 ).

0.01

0.1

1

10

-10 -5 0 5

F(y)

y

F(y)

e-y

FIGURE 4. F(y) behavior featuring function maximum and long-term trending.

General Solution Approach

The approach used to identify functions such as F(y) and G(η, x) may be formulated 

in a more general sense beginning with Eq. (2). Casting Eq. (1) in terms of s, the forcing 

term may be introduced as 

( ) f(s)

c

q s = , (16)
7

leading to

( ) ( ) ( ) ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ − ∫ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = + ∫ −Δ − −Δ − 1

1

0 0

1

s u s s s u s s du

f u

e du e e

f u

e

p t p e c e . (17)

If it is possible to introduce the variable substitution dξ = du f(u) such that u = g(ξ )

is a function in terms of the transformed variable, then the scaled integral will be

( )

( ) ( ) ξ (ξ ) ξ ξ du e e d H

f u

e

e g g

s u s ∫ ∝ ∫ ≡ − −

0

. (18)

Function H(ξ) will satisfy its own differential equation:

+ = 1

ξ dξ

dg H

d

dH . (19)

If well-behaved, it appears these functions should all rise from some initial 

transformed variable zero with a slope of unity, and for large arguments will approach 

limiting behavior of

( ) (ξ ) ξ '

1

g

H → ∞ → . (20) 

H(ξ) will also form the core of the solution to integrals of the type arrived at in Eq. (17).

Concluding Remarks

A new mathematical function G(η, x) has been discovered that solves integrals 

containing the form u e du −η u through a simple transformation from u to x. The integral 

is solved through use of a property of G as a solution to a particular differential equation. 

This function is useful for describing transient behavior for pressure in vacuum systems, 

and several properties of this function have been explored. 

It has been noted that Dawson’s function is related to certain solutions of the 

diffusion equation used in conduction heat transfer and mass transfer studies, as well as 

certain investigations associated with propagation of electromagnetic radiation. Since 

G(η, x) simplifies to Dawson’s function for η = 0.5, it seems logical that the new function 

will offer at least the same level of utility.

The special case η = 1 cannot be solved with the transformation associated with 

development of G, but may be handled by a similar approach. The resulting function 

F(ln u) is related to the exponential integral.
8

Finally, the approach taken to generate these functions has been generalized to 

identify properties of solutions to integrals containing the form (e f(u))du u . Provided 

certain conditions are met, these solutions will share certain features of short-period and 

long-period behavior.

References

1. Woronowicz, M., “On small disturbance ascent vent behavior,” in Optical System 

Contamination: Effects, Measurements, and Control 2012, eds. S. Straka, et al., 

Proceedings of SPIE Vol. 8492 (SPIE, Bellingham, WA, 2012), 8492-13.

2. Bird, R., et al., Transport Phenomena, John Wiley & Sons, New York, 1960, pp. 

555-601.

3. Redhead, P., “Modeling the pump-down of a reversibly adsorbed phase. I. 

Monolayer and submonolayer initial coverage,” Journal of Vacuum Science & 

Technology A,.13, 1995, pp. 467-475.

4. Huang, A., et al., “Materials Outgassing Rate Decay in Vacuum at Isothermal 

Conditions,” in Systems Contamination: Prediction, Control, and Performance 2016,

eds. N. Carosso and J. Egges, Proceedings of SPIE Vol. 9952 (SPIE, Bellingham, 

WA, 2016), 9952-06.

5. Gradshteyn, I., and Ryzhik, I., Table of Integrals, Series, and Products, Corrected 

and Enlarged Edition, Academic Press, San Francisco, 1980, 2.324-2 & 2.325, pp. 

92-3.

6. Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical 

Tables, eds. M. Abramowitz and I. Stegun, National Bureau of Standards, Applied 

Mathematics Series 55, Washington, DC, June 1964, pp. 228-237.

7. Oldham, K., et al., An Atlas of Functions, 2nd ed., Springer Science+Business Media, 

LLC, New York, pp. 375-83, 2009.

8. Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical 

Tables, eds. M. Abramowitz and I. Stegun, National Bureau of Standards, Applied 

Mathematics Series 55, Washington, DC, June 1964, pp. 297-319.

9. Oldham, K., et al., An Atlas of Functions, 2nd ed., Springer Science+Business Media, 

LLC, New York, pp. 427-33, 2009.

10. ibid., pp. 461-70.

11. Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical 

Tables, eds. M. Abramowitz and I. Stegun, National Bureau of Standards, Applied 

Mathematics Series 55, Washington, DC, June 1964, pp. 260-263.

12. Temme, N., “Computational Aspects of Incomplete Gamma Functions with Large 

Complex Parameters,” International Series of Numerical Mathematics, 119, 1994, 

pp. 551-562.