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[[File:Improperintegral2.png|right|thumb|200px|An improper integral of the first kind. The integral may need to be defined on an unbounded domain.]]
{{Infobox scientist
[[File:Improperintegral1.png|right|thumb|200px|An improper Riemann integral of the second kind. The integral may fail to exist because of a [[vertical asymptote]] in the function.]]
|name              = Ronald N. Bracewell
{{Calculus |Integral}}
|image            =
|image_size      =
|caption          =
|birth_date        = {{Birth date|1921|07|22}}
|birth_place      = [[Sydney, New South Wales|Sydney]], [[New South Wales]], [[Australia]]
|death_date        = {{Death date and age|2007|08|12|1921|07|22}}
|death_place      = [[Stanford, California|Stanford]], [[California]], [[United States of America|USA]]
|residence        =
|citizenship      =
|nationality      = [[Australia]]n
|ethnicity        =
|field            = [[Physics|Physicist]]<br />[[Mathematics|Mathematician]]<br />[[Radio astronomy|Radio astronomer]]
|work_institutions = [[Commonwealth Scientific and Industrial Research Organisation|CSIRO]]<br />[[University of California, Berkeley]]<br />[[Stanford University]]
|alma_mater        = [[University of Sydney]]<br />[[University of Cambridge]] [[Sydney Boys High School]]
|doctoral_advisor  =
|doctoral_students =
|known_for        =
|author_abbrev_bot =
|author_abbrev_zoo =
|influences        =
|influenced        =
|prizes            = [[IEEE Heinrich Hertz Medal]] (1994)<br />Officer of the [[Order of Australia]] (1998)
|religion          =
|footnotes        =
}}
'''Ronald Newbold Bracewell''' [[Order of Australia|AO]] (July 22, 1921 &ndash; August 12, 2007) was the [[Lewis M. Terman]] Professor of Electrical Engineering, Emeritus of the Space, Telecommunications and Radioscience Laboratory at [[Stanford University]].


In [[calculus]], an '''improper integral''' is the [[limit (mathematics)|limit]] of a [[definite integral]] as an endpoint of the interval(s) of integration approaches either a specified [[real number]] or ∞ or &minus;∞ or, in some cases, as both endpoints approach limits. Such an integral is often written symbolically just like a standard definite integral, perhaps with ''infinity'' as a limit of integration. But that conceals the limiting process. By using the more advanced [[Lebesgue integral]], rather than the [[Riemann integral]], one can in some cases get an answer without taking a limit of standard definite integrals.
== Education ==


Specifically, an improper integral is a limit of the form
Bracewell was born in [[Sydney]], [[Australia]], in 1921, and educated at [[Sydney Boys High School]]. He graduated from the [[University of Sydney]] in 1941 with the B.Sc. degree in mathematics and physics, later receiving the degrees of B.E. (1943), and M.E. (1948) with first class honours, and while working in the Engineering Department became the President of the Oxometrical Society.  During [[World War II]] he designed and developed microwave radar equipment in the Radiophysics Laboratory of the Commonwealth Scientific and Industrial Research Organisation, Sydney under the direction of [[Joseph Lade Pawsey|Joseph L. Pawsey]] and [[Edward George Bowen|Edward G. Bowen]] and from 1946 to 1949 was a research student at [[Sidney Sussex College]], [[Cambridge]], engaged in [[ionosphere|ionospheric]] research in the [[Cavendish Laboratory]], where he received his Ph.D. degree in physics under [[J. A. Ratcliffe]].
:<math>\lim_{b\to\infty} \int_a^bf(x)\, \mathrm{d}x, \qquad \lim_{a\to -\infty} \int_a^bf(x)\, \mathrm{d}x,</math>
or of the form
:<math>\lim_{c\to b^-} \int_a^cf(x)\, \mathrm{d}x,\quad
\lim_{c\to a^+} \int_c^bf(x)\, \mathrm{d}x,</math>
in which one takes a limit in one or the other (or sometimes both) endpoints {{harv|Apostol|1967|loc=§10.23}}. Integrals are also improper if the integrand is undefined at an [[interior point]] of the domain of integration, or at multiple such points.


It is often necessary to use improper integrals in order to compute a value for integrals which may not exist in the conventional sense (as a [[Riemann integral]], for instance) because of a singularity in the function, or an infinite endpoint of the domain of integration.
== Career ==


==Examples==
From October 1949 to September 1954 Dr. Bracewell was a Senior Research Officer at the Radiophysics Laboratory of the [[CSIRO]], Sydney, concerned with very long wave propagation and [[radio astronomy]]. He then lectured in radio astronomy at the Astronomy Department of the [[University of California, Berkeley]] from September 1954 to June 1955 at the invitation of [[Otto Struve]], and at Stanford University during the summer of 1955, and joined the Electrical Engineering faculty at Stanford in December 1955.
The original definition of the [[Riemann integral]] does not apply to a function such as <math>1/{x^2}</math> on the interval [1, ∞), because in this case the domain of integration is [[bounded set|unbounded]]. However, the Riemann integral can often be extended by [[continuous function|continuity]], by defining the improper integral instead as a [[Limit (mathematics)|limit]]


:<math>\int_1^\infty \frac{1}{x^2}\,\mathrm{d}x=\lim_{b\to\infty} \int_1^b\frac{1}{x^2}\,\mathrm{d}x = \lim_{b\to\infty} \left(-\frac{1}{b} + \frac{1}{1}\right) = 1. </math>
In 1974 he was appointed the first Lewis M. Terman Professor and Fellow in Electrical Engineering (1974–1979).  Though he retired in 1979, he continued to be active until his death.


The narrow definition of the Riemann integral also does not cover the function <math>1/\sqrt{x}</math> on the interval [0, 1]. The problem here is that the integrand is [[bounded function|unbounded]] in the domain of integration (the definition requires that both the domain of integration and the integrand be bounded). However, the improper integral does exist if understood as the limit
== Contributions and honours ==


:<math>\int_0^1 \frac{1}{\sqrt{x}}\,\mathrm{d}x=\lim_{a\to 0^+}\int_a^1\frac{1}{\sqrt{x}}\, \mathrm{d}x = \lim_{a\to 0^+}(2\sqrt{1}-2\sqrt{a})=2.</math>
Professor Bracewell was a Fellow of the [[Royal Astronomical Society]] (1950), Fellow and life member of the [[Institute of Electrical and Electronic Engineers]] (1961), Fellow of the [[American Association for the Advancement of Science]] (1989), and was a Fellow with other significant societies and organisations.


== Convergence of the integral ==
For experimental contributions to the study of the ionosphere by means of very low frequency waves, Dr. Bracewell received the  Duddell Premium of the Institution of Electrical Engineers, London in 1952. In 1992 he was elected to foreign associate membership of the Institute of Medicine of the U.S. [[United States National Academy of Sciences|National Academy of Sciences]] (1992), the first Australian to achieve that distinction, for fundamental contributions to medical imaging. He was one of Sydney University's three honourees when alumni awards were instituted in 1992, with a citation for brain scanning, and was the 1994 recipient of the Institute of Electrical and Electronic Engineers' Heinrich Hertz medal for pioneering work in antenna [[aperture synthesis]] and image reconstruction as applied to radio astronomy and to computer-assisted tomography. In 1998 Dr. Bracewell was named Officer of the [[Order of Australia]] (AO) for service to science in the fields of radio astronomy and image reconstruction.
An improper integral converges if the limit defining it exists. Thus for example one says that the improper integral
:<math>\lim_{t\to\infty} \int_a^t f(x)\, \mathrm{d}x</math>
exists and is equal to ''L'' if the integrals under the limit exist for all sufficiently large ''t'', and the value of the limit is equal to ''L''.


It is also possible for an improper integral to diverge to infinity. In that case, one may assign the value of ∞ (or &minus;∞) to the integral.  For instance
At CSIRO Radiophysics Laboratory, work that in 1942-1945 was classified appeared in a dozen reports. Activities included design, construction, and demonstration of voice-modulation equipment for a 10&nbsp;cm magnetron (July 1943), a microwave triode oscillator at 25&nbsp;cm using cylindrical cavity resonators, equipment designed for microwave radar in field use (wavemeter, echo box, thermistor power meter, etc.) and microwave measurement technique. Experience with numerical computation of fields in cavities led, after the war, to a Master of Engineering degree (1948) and the definitive publication on step discontinuities in radial transmission lines (1954).
:<math>\lim_{b\to\infty}\int_1^b \frac{1}{x}\,\mathrm{d}x = \infty.</math>
However, other improper integrals may simply diverge in no particular direction, such as
:<math>\lim_{b\to\infty}\int_1^b x\sin x\, \mathrm{d}x,</math>
which does not exist, even as an [[extended real number]]. This is called divergence by oscillation.


A limitation of the technique of improper integration is that the limit must be taken with respect to one endpoint at a time.  Thus, for instance, an improper integral of the form
While at the Cavendish Laboratory, Cambridge (1946–1950) Bracewell worked on observation and theory of upper atmospheric ionisation, contributing to experimental technique (1948), explaining solar effects (1949), and distinguishing two layers below the E-layer (1952), work recognised by the Duddell Premium.


:<math>\int_{-\infty}^\infty f(x)\, \mathrm{d}x</math>
At Stanford Professor Bracewell constructed a microwave spectroheliograph (1961), a large and complex radio telescope which produced daily temperature maps of the sun reliably for eleven years, the duration of a solar cycle.  The first [[radio telescope]] to give output automatically in printed form, and therefore capable of worldwide dissemination by teleprinter, its daily solar weather maps received acknowledgement from [[NASA]] for support of the first manned landing on the moon.


can be defined by taking two separate limits; to wit
Many fundamental papers on restoration (1954–1962), [[interferometry]] (1958–1974) and reconstruction (1956–1961) appeared along with instrumental and observational papers.  By 1961 the radio-interferometer calibration techniques developed for the [[spectroheliograph]] first allowed an antenna system, with 52" fan beam, to equal the angular resolution of the human eye in one observation.  With this beam the components of [[Cygnus A]], spaced 100", were put directly in evidence without the need for repeated observations with variable spacing [[aperture synthesis]] interferometry.


:<math>\int_{-\infty}^\infty f(x)\, \mathrm{d}x = \lim_{a\to -\infty} \lim_{b\to \infty} \int_a^bf(x) \, \mathrm{d}x</math>
The nucleus of the extragalactic source [[Centaurus A]] was resolved into two separate components whose right ascensions were accurately determined with a 2.3-minute fan beam at 9.1&nbsp;cm.  Knowing that Centaurus A was composite, Bracewell used the 6.7-minute beam of the [[Parkes Observatory]] 64 m [[radiotelescope]] at 10&nbsp;cm to determine the separate declinations of the components and in so doing was the first to observe strong polarisation in an extragalactic source (1962), a discovery of fundamental significance for the structure and role of astrophysical magnetic fields. Subsequent observations made at Parkes by other observers with a 14-minute and wider beams at [[21 cm line|21 cm]] and longer wavelengths, though not resolving the components, were compatible with the <math>\lambda^2</math> dependence expected from Faraday rotation if magnetic fields were the polarising agent.


provided the double limit is finite. It can also be defined as a pair of distinct improper integrals of the first kind:
A second major radiotelescope (1971) employing advanced concepts to achieve an angular resolution of 18 seconds of arc was designed and built at Stanford and applied to both solar and galactic studies. The calibration techniques for this leading-edge resolution passed into general use in radio interferometry via the medium of alumni.


:<math>\lim_{a\to -\infty}\int_a^cf(x)\, \mathrm{d}x + \lim_{b\to \infty} \int_c^b f(x) \, \mathrm{d}x</math>
Upon the discovery of the [[cosmic microwave background radiation|cosmic background radiation]]:
* a remarkable observational limit of 1.7 millikelvins, with considerable theoretical significance for cosmology, was set on the anisotropy in collaboration with Ph. D. student [[E.K. Conklin]] (1967), and was not improved on for many years
* the correct theory of a relativistic observer in a blackbody enclosure (1968) was given in the first of several papers by various authors obtaining the same result
* the absolute motion of the [[Sun]] at 308&nbsp;km/s through the cosmic background radiation was measured by Conklin in 1969, some years before independent confirmation.


where ''c'' is any convenient point at which to start the integration. This definition also applies when one of these integrals is infinite, or both if they have the same sign.
With the advent of the space age, Bracewell became interested in [[celestial mechanics]], made observations of the radio emission from [[Sputnik 1]], and supplied the press with accurate charts predicting the path of Soviet satellites, which were perfectly visible, if you knew when and where to look. Following the puzzling performance of [[Explorer I]] in orbit, he published the first explanation (1958-9) of the observed spin instability of satellites, in terms of the Poinsot motion of a non-rigid body with internal friction. He recorded the signals from Sputniks I, II and III and discussed them in terms of the satellite spin, antenna polarisation, and propagation effects of the ionised medium, especially Faraday effect.


An example of an improper integrals where both endpoints are infinite is the [[Gaussian integral]] <math>\int_{-\infty}^\infty e^{-x^2}\,\mathrm{d}x = \sqrt{\pi}</math>. An example which evaluates to infinity is <math>\int_{-\infty}^\infty e^{x}\,\mathrm{d}x</math>. But one cannot even define other integrals of this kind unambiguously, such as <math>\int_{-\infty}^\infty x\,\mathrm{d}x</math>, since the double limit is infinite and the two-integral method
Later (1978, 1979) he invented a spinning, [[Nuller|nulling]], two-element infrared interferometer suitable for space-shuttle launching into an orbit near [[Jupiter]], with milliarcsecond resolution, that could lead to the discovery of [[Extrasolar planet|planets around stars other than the sun]]. This concept was elaborated in 1995 by Angel and Woolf, whose space-station version with four-element double nulling became the [[Terrestrial Planet Finder]] (TPF), NASA's candidate for imaging planetary configurations of other stars.<ref>''Scientific American'', April 1996</ref>


:<math>\lim_{a\to -\infty}\int_a^cx\,\mathrm{d}x + \lim_{b\to\infty} \int_c^b x\,\mathrm{d}x</math>
Imaging in astronomy led to participation in development of computer assisted x-ray tomography, where commercial scanners reconstruct tomographic images using the algorithm developed by Bracewell for radioastronomical reconstruction from fan-beam scans. This corpus of work has been recognized by the Institute of Medicine, an award by the [[University of Sydney]], and the Heinrich Hertz medal. Service on the founding editorial board of the ''Journal for Computer-Assisted Tomography'', to which he also contributed publications, and on the scientific advisory boards of medical instrumentation companies maintained Bracewell's interest in medical imaging, which became an important part of his regular graduate lectures on imaging, and forms an important part of his 1995 text on imaging.
yields <math>\infty-\infty</math>. In this case, one can however define an improper integral in the sense of [[Cauchy principal value]]:


:<math> \operatorname{p.v.} \int_{-\infty}^\infty x\,\mathrm{d}x = \lim_{b\to\infty}\int_{-b}^b x \, \mathrm{d}x = 0.</math>
Experience with the optics, mechanics and control of radiotelescopes led to involvement with solar thermophotovoltaic energy at the time of the energy crisis, including the fabrication of low-cost solid and perforated paraboloidal reflectors by hydraulic inflation.


The questions one must address in determining an improper integral are:
Bracewell is also known for being the first to propose the use of autonomous [[interstellar]] [[space probes]] for communication between alien civilisations as an alternative to [[radio]] transmission dialogs. This hypothetical concept has been dubbed the [[Bracewell probe]] after its inventor.


*Does the limit exist?
=== Fourier analysis ===
*Can the limit be computed?


The first question is an issue of [[mathematical analysis]]. The second one can be addressed by calculus techniques, but also in some cases by [[contour integration]], [[Fourier transform]]s and other more advanced methods.
As a consequence of relating images to [[Fourier analysis]], in 1983 he discovered a [[Discrete Hartley transform|new factorisation of the discrete Fourier transform]] matrix leading to a fast algorithm for spectral analysis. This method, which has advantages over the fast Fourier algorithm, especially for images, is treated in [[Hartley transform|The Hartley Transform]] (1986), in U.S. Patent 4,646,256 (1987, now in the public domain), and in over 200 technical papers by various authors that were stimulated by the discovery. Analogue methods of creating a Hartley transform plane first with light and later with microwaves were demonstrated in the laboratory and permitted the determination of electromagnetic phase by the use of square-law detectors.  A new elementary signal representation, the [[Chirplet transform]], was discovered (1991) that complements the Gabor elementary signal representations used in dynamic spectral analysis (with the property of meeting the bandwidth-duration minimum associated with the [[uncertainty principle]]). This advance opened a new field of adaptive dynamic spectra with wide application in information analysis.


==Types of integrals==
=== Other interests ===
There is more than one theory of [[integral|integration]]. From the point of view of calculus, the [[Riemann integral]] theory is usually assumed as the default theory. In using improper integrals, it can matter which integration theory is in play.


* For the Riemann integral (or the [[Darboux integral]], which is equivalent to it), improper integration is necessary ''both'' for unbounded intervals (since one cannot divide the interval into finitely many subintervals of finite length) ''and'' for unbounded functions with finite integral (since, supposing it is unbounded above, then the upper integral will be infinite, but the lower integral will be finite).
Professor Bracewell was interested in conveying an appreciation of the role of science in society to the public, in mitigating the effects of scientific illiteracy on public decision making through contact with alumni groups, and in liberal undergraduate education within the framework of the Astronomy Course Program and the Western Culture program in Values, Technology, Science and Society, in both of which he taught for some years. He gave the 1996 Bunyan Lecture on ''The Destiny of Man''.
* The [[Lebesgue integral]] deals differently with unbounded domains and unbounded functions, so that often an integral which only exists as an improper Riemann integral will exist as a (proper) Lebesgue integral, such as <math>\int_1^\infty \frac{1}{x^2}\,\mathrm{d}x</math>.  On the other hand, there are also integrals that have an improper Riemann integral but do not have a (proper) Lebesgue integral, such as <math>\int_0^\infty \frac{\sin x}{x}\,\mathrm{d}x</math>. The Lebesgue theory does not see this as a deficiency: from the point of view of [[measure theory]], <math>\int_0^\infty \frac{\sin x}{x}\,\mathrm{d}x = \infty - \infty</math> and cannot be defined satisfactorily.  In some situations, however, it may be convenient to employ improper Lebesgue integrals as is the case, for instance, when defining the [[Cauchy principal value]]. The Lebesgue integral is more or less essential in the theoretical treatment of the [[Fourier transform]], with pervasive use of integrals over the whole real line.
* For the [[Henstock–Kurzweil integral]], improper integration ''is not necessary'', and this is seen as a strength of the theory: it encompasses all Lebesgue integrable and improper Riemann integrable functions.


== Improper Riemann integrals and Lebesgue integrals ==
He was also interested in the trees of Stanford's campus and published a book about them. He also taught an undergraduate seminar titled I Dig Trees.<ref>{{cite web|url=http://trees.stanford.edu/ |title=Trees of Stanford |publisher=Trees.stanford.edu |date= |accessdate=2012-04-19}}</ref><ref>{{cite web|url=http://news-service.stanford.edu/news/2005/march30/trees-033005.html |title=More than 350 tree species on campus cataloged by professor in new book |publisher=News-service.stanford.edu |date=2005-03-30 |accessdate=2012-04-19}}</ref>
[[File:Improperintegral1.png|right|thumb|200px|Figure 1]]
[[File:Improperintegral2.png|right|thumb|200px|Figure 2]]


In some cases, the integral
Bracewell was also a designer and builder of [[sundial]]s. He built one on the South side of the Terman Engineering Building. He built one at the home of his son, Mark Bracewell. He built another on the deck of professor John Linvill's house.


:<math>\int_a^c f(x)\,\mathrm{d}x\,</math>
As his seminar "I Dig Trees" indicated, Dr. Bracewell was known for having a tremendously keen, intelligent sense of wry, science-infused humor.  One of his treasured family photos showed him sitting on the ground, legs akimbo, with a beer bottle in front of him that he had neatly balanced on one of its bottom edges—his proof that even that thin edge had 3 balance points.


can be defined as an integral (a [[Lebesgue integral]], for instance) without reference to the limit
==Selected publications==


:<math>\lim_{b\to c^-}\int_a^b f(x)\,\mathrm{d}x\,</math>
*Bracewell, R.N. and Pawsey, J.L., ''Radio Astronomy'' (Oxford, 1955) ''(also translated into Russian and reprinted in China)''
*Bracewell, R.N., ''Radio Interferometry of Discrete Sources'' (Proceedings of the IRE, January 1958)
*Bracewell, Ronald N., ed., ''Paris Symposium on Radio Astronomy, IAU Symposium no. 9 and URSI Symposium no. 1, held 30 July 1958 &ndash; 6 August 1958'' (Stanford Univ. Press, Stanford, CA, 1959) ''(also translated into Russian)''
*Professor Bracewell translated ''Radio Astronomy'', by J.L. Steinberg and J. Lequeux, (McGraw-Hill, 1963) from French
*Bracewell, R.N., ''The Fourier Transform and Its Applications'' (McGraw-Hill, 1965, 2nd ed. 1978, revised 1986) ''(also translated into Japanese and Polish)''
*Bracewell, R.N., ''Trees on the Stanford Campus'' (Stanford: Samizdat, 1973)
*Bracewell, R.N., ''The Galactic Club: Intelligent Life in Outer Space'' (Portable Stanford: Alumni Association, 1974) ''(also translated into Dutch, Japanese, and Italian)''
*Bracewell, R.N., ''The Hartley Transform'' (Oxford University Press, 1986) ''(also translated into German and Russian)''
*Bracewell, R.N., ''Two-Dimensional Imaging'' (Prentice-Hall, 1995)
*Bracewell, R.N., ''Fourier Analysis and Imaging'' (Plenum, 2004)


but cannot otherwise be conveniently computed. This often happens when the function ''f'' being integrated from ''a'' to ''c'' has a [[vertical asymptote]] at ''c'', or if ''c''&nbsp;=&nbsp;∞ (see Figures 1 and 2).  In such cases, the improper Riemann integral allows one to calculate the Lebesgue integral of the function.  Specifically, the following theorem holds  {{harv|Apostol|1974|loc=Theorem 10.33}}:
*Bracewell, R.N., ''Trees of Stanford and Environs'' (Stanford Historical Society, 2005)


* If a function ''f'' is Riemann integrable on [''a'',''b''] for every ''b''&nbsp;≥&nbsp;''a'', and the partial integrals
==Chapter contributions==
::<math>\int_a^b|f(x)|\,\mathrm{d}x</math>
Bracewell has contributed chapters to:
:are bounded as ''b''&nbsp;&rarr;&nbsp;∞, then the improper Riemann integrals
::<math>\int_a^\infty f(x)\, \mathrm{d}x,\quad\mbox{and}\ \int_a^\infty |f(x)|\, \mathrm{d}x</math>
:both exist.  Furthermore, ''f'' is Lebesgue integrable on [''a'', ∞), and its Lebesgue integral is equal to its improper Riemann integral.


For example, the integral
*Textbook of Radar ''Microwave Transmission and Cavity Resonator Theory,'' ed. E.G. Bowen, 1946
:<math>\int_0^\infty\frac{\mathrm{d}x}{1+x^2}</math>
*Advances in Astronautical Sciences ''Satellite Rotation,'' ed.  H. Jacobs, 1959
can be interpreted alternatively as the improper integral
*The Radio Noise Spectrum ''Correcting Noise Maps for Beamwidth,'' ed. D.H. Menzel, 1960
:<math>\lim_{b\to\infty}\int_0^b\frac{\mathrm{d}x}{1+x^2}=\lim_{b\to\infty}\arctan{b}=\frac{\pi}{2},</math>
*Modern Physics for the Engineer ''Radio Astronomy,'' ed. L. Ridenour and [[William Nierenberg|W. Nierenberg]], 1960
or it may be interpreted instead as a [[Lebesgue integral]] over the set (0, ∞). Since both of these kinds of integral agree, one is free to choose the first method to calculate the value of the integral, even if one ultimately wishes to regard it as a Lebesgue integral. Thus improper integrals are clearly useful tools for obtaining the actual values of integrals.
*Statistical methods in Radio Wave Propagation ''Antenna Tolerance Theory,'' ed. W.C. Hoffman, 1960
*Advances in Geophysics ''Satellite Studies of the Ionization in Space by Radio,'' ed. H.E. Landsberg, 1961 (O.K. Garriott and R.N.  Bracewell)
*Handbuch der Physik ''Radio Astronomy Techniques,'' ed. S. Flugge, 1962
*Encyclopedia of Electronics ''Extraterrestrial Radio Noise'', ed. C. Susskind, 1962
*Stars and Galaxies ''Radio Broadcasts from the Depths of Space,'' ed. T.L Page, 1962
*Radio Waves and Circuits ''Aerials and Data Processing,'' ed. S. Silver, 1963
*Light and Life in the Universe ''Life in the Galaxy,'' ed. S.T. Butler and H. Messel, 1964
*Encyclopædia Britannica ''Telescope, Radio'', 1967
*Vistas in Science ''The Microwave Sky,'' ed. David L. Arm, 1968
*Man in Inner and Outer Space ''The Sun (Five Chapters),'' ed. H. Messel and S.T. Butler, 1968
*Image Reconstruction from Projections: Implementation and Applications ''Image Reconstruction in Radio Astronomy,'' ed. G. Hermann, 1979
*Annual Review of Astronomy and Astrophysics ''Computer Image Processing,'' ed. G. Burbidge et al., 1979
*Energy for Survival ''How It All Began,'' ''Man the Lazy Animal,'' and ''Energy from Sunlight,'' ed. H. Messel, 1979
*Life in the Universe ''Manifestations of Advanced Civilizations,'' ed. J. Billingham, 1981
*Extraterrestrials: Where Are They? ''Preemption of the Galaxy by the First Advanced Civilization,'' ed. M.H. Hart and B. Zuckerman, 1982, 1995
*Transformations in Optical Signal Processing ''Fourier Inversion of Deficient Data,'' ed. W.T. Rhodes, J.R. Fienup and B.E.A. Saleh, 1984
*The Early Years of Radio Astronomy ''Early Work on Imaging Theory in Radio Astronomy,'' ed. W.T. Sullivan, III, 1984
*Indirect Imaging ''Inversion of Nonplanar Visibilities,'' ed. J.A. Roberts, 1984
*Fourier Techniques and Applications ''The Life of Joseph Fourier'' and ''Fourier Techniques and Applications,'' ed. J.F. Price, 1985
*Yearbook of Science and Technology ''Wavelets,'' 1996
*Encyclopedia of Applied Physics ''Fourier and Other Mathematical Transforms'' 1997
*[[Cornelius Lanczos]]&mdash;Collected Published Papers with Commentaries ''The Fast Fourier Transform'' and''Smoothing Data by Analysis and by Eye'' ed. W.R. Davis et al., 1999


In other cases, however, a Lebesgue integral between finite endpoints may not even be defined, because the integrals of the positive and negative parts of ''f'' are both infinite, but the improper Riemann integral may still exist.  Such cases are "properly improper" integrals, i.e. their values cannot be defined except as such limits.  For example,
==References==
{{reflist}}


:<math>\int_0^\infty\frac{\sin(x)}{x}\,\mathrm{d}x</math>
==External links==
 
*[http://www-star.stanford.edu/people/bracewell.html Stanford Web Profile]
cannot be interpreted as a Lebesgue integral, since
*[http://news-service.stanford.edu/pr/2007/pr-bracewell-082207.html Stanford Death Notice]
 
:<math>\int_0^\infty\left|\frac{\sin(x)}{x}\right|\,\mathrm{d}x=\infty.</math>
 
But <math>f(x)=\sin(x)/x</math> is nevertheless Riemann integrable between any two finite endpoints, and its integral between 0 and ∞ is usually understood as the limit of the Riemann integral:
 
:<math>\int_0^\infty\frac{\sin(x)}{x}\,\mathrm{d}x=\lim_{b\rightarrow\infty}\int_0^b\frac{\sin(x)}{x}\,\mathrm{d}x=\frac{\pi}{2}.</math>
 
==Singularities==
 
One can speak of the ''singularities'' of an improper integral, meaning those points of the [[extended real number line]] at which limits are used.
 
==Cauchy principal value==
{{main|Cauchy principal value}}
Consider the difference in values of two limits:
 
:<math>\lim_{a\rightarrow 0+}\left(\int_{-1}^{-a}\frac{\mathrm{d}x}{x}+\int_a^1\frac{\mathrm{d}x}{x}\right)=0,</math>
 
:<math>\lim_{a\rightarrow 0+}\left(\int_{-1}^{-a}\frac{\mathrm{d}x}{x}+\int_{2a}^1\frac{\mathrm{d}x}{x}\right)=-\ln 2.</math>
 
The former is the Cauchy principal value of the otherwise ill-defined expression
 
:<math>\int_{-1}^1\frac{\mathrm{d}x}{x}{\  }
\left(\mbox{which}\  \mbox{gives}\  -\infty+\infty\right).</math>
 
Similarly, we have
 
:<math>\lim_{a\rightarrow\infty}\int_{-a}^a\frac{2x\,\mathrm{d}x}{x^2+1}=0,</math>
 
but
 
:<math>\lim_{a\rightarrow\infty}\int_{-2a}^a\frac{2x\,\mathrm{d}x}{x^2+1}=-\ln 4.</math>
 
The former is the principal value of the otherwise ill-defined expression
 
:<math>\int_{-\infty}^\infty\frac{2x\,\mathrm{d}x}{x^2+1}{\  }
\left(\mbox{which}\  \mbox{gives}\  -\infty+\infty\right).</math>
 
All of the above limits are cases of the [[indeterminate form]] ∞ &minus; ∞.
 
These [[pathological (mathematics)|pathologies]] do not affect "Lebesgue-integrable" functions, that is, functions the integrals of whose [[absolute value]]s are finite.
 
==Summability==
An indefinite integral may diverge in the sense that the limit defining it may not exist.  In this case, there are more sophisticated definitions of the limit which can produce a convergent value for the improper integral.  These are called [[summability]] methods.
 
One summability method, popular in [[Fourier analysis]], is that of [[Cesàro summation]].  The integral
 
:<math>\int_0^\infty f(x)\,\mathrm{d}x</math>
 
is Cesàro summable (C,&nbsp;α) if
 
:<math>\lim_{\lambda\to\infty}\int_0^\lambda\left(1-\frac{x}{\lambda}\right)^\alpha f(x)\, \mathrm{d}x </math>
 
exists and is finite {{harv|Titchmarsh|1948|loc=§1.15}}. The value of this limit, should it exist, is the (C,&nbsp;α) sum of the integral.
 
An integral is (C,&nbsp;0) summable precisely when it exists as an improper integral.  However, there are integrals which are (C,&nbsp;α) summable for α&nbsp;>&nbsp;0 which fail to converge as improper integrals (in the sense of Riemann or Lebesgue). One example is the integral
 
:<math>\int_0^\infty\sin x\, \mathrm{d}x</math>
 
which fails to exist as an improper integral, but is (C,α) summable for every α&nbsp;>&nbsp;0. This is an integral version of [[Grandi's series]].


==Bibliography==
{{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
* {{citation|last=Apostol|first=T|authorlink=Tom M. Apostol|title=Mathematical analysis|publisher=Addison-Wesley|year=1974|isbn=978-0-201-00288-1}}.
| NAME              = Bracewell, Ronald
* {{citation|last=Apostol|first=T|authorlink=Tom M. Apostol|title=Calculus, Vol. 1|publisher=Jon Wiley & Sons|edition=2nd|year=1967}}.
| ALTERNATIVE NAMES =  
*{{Citation
| SHORT DESCRIPTION =  
|author=Autar Kaw, Egwu Kalu
| DATE OF BIRTH    = 1921-07-22
|year=2008
| PLACE OF BIRTH    = [[Sydney, New South Wales|Sydney]], [[New South Wales]], [[Australia]]
|title=Numerical Methods with Applications
| DATE OF DEATH    = 2007-08-12
|url=http://numericalmethods.eng.usf.edu/topics/textbook_index.html
| PLACE OF DEATH    = [[Stanford, California|Stanford]], [[California]], [[United States of America|USA]]
|edition=1st
|publisher=autarkaw.com
|isbn=
}}
}}
* {{citation|last=Titchmarsh|first=E|authorlink=Edward Charles Titchmarsh|title=Introduction to the theory of Fourier integrals|isbn=978-0-8284-0324-5|year=1948|edition=2nd|publication-date=1986|publisher=Chelsea Pub. Co.|location=New York, N.Y.}}.
{{DEFAULTSORT:Bracewell, Ronald}}
 
[[Category:1921 births]]
==External links==
[[Category:2007 deaths]]
* [http://numericalmethods.eng.usf.edu/topics/improper_integration.html Numerical Methods to Solve Improper Integrals] at Holistic Numerical Methods Institute
[[Category:Stanford University School of Engineering faculty]]
* [http://www.lightandmatter.com/html_books/calc/ch06/ch06.html Improper integrals] – chapter from an online textbook
[[Category:Australian astronomers]]
[[Category:Australian physicists]]
[[Category:Officers of the Order of Australia]]
[[Category:Fellows of St Catherine's College, Oxford]]
[[Category:Fellows of the American Association for the Advancement of Science]]
[[Category:Fellow Members of the IEEE]]
[[Category:20th-century astronomers]]
[[Category:Alumni of Sidney Sussex College, Cambridge]]
[[Category:Search for extraterrestrial intelligence]]
[[Category:People educated at Sydney Boys High School]]


[[Category:Calculus]]
[[fr:Ronald Bracewell]]

Revision as of 13:12, 11 August 2014

Template:Infobox scientist Ronald Newbold Bracewell AO (July 22, 1921 – August 12, 2007) was the Lewis M. Terman Professor of Electrical Engineering, Emeritus of the Space, Telecommunications and Radioscience Laboratory at Stanford University.

Education

Bracewell was born in Sydney, Australia, in 1921, and educated at Sydney Boys High School. He graduated from the University of Sydney in 1941 with the B.Sc. degree in mathematics and physics, later receiving the degrees of B.E. (1943), and M.E. (1948) with first class honours, and while working in the Engineering Department became the President of the Oxometrical Society. During World War II he designed and developed microwave radar equipment in the Radiophysics Laboratory of the Commonwealth Scientific and Industrial Research Organisation, Sydney under the direction of Joseph L. Pawsey and Edward G. Bowen and from 1946 to 1949 was a research student at Sidney Sussex College, Cambridge, engaged in ionospheric research in the Cavendish Laboratory, where he received his Ph.D. degree in physics under J. A. Ratcliffe.

Career

From October 1949 to September 1954 Dr. Bracewell was a Senior Research Officer at the Radiophysics Laboratory of the CSIRO, Sydney, concerned with very long wave propagation and radio astronomy. He then lectured in radio astronomy at the Astronomy Department of the University of California, Berkeley from September 1954 to June 1955 at the invitation of Otto Struve, and at Stanford University during the summer of 1955, and joined the Electrical Engineering faculty at Stanford in December 1955.

In 1974 he was appointed the first Lewis M. Terman Professor and Fellow in Electrical Engineering (1974–1979). Though he retired in 1979, he continued to be active until his death.

Contributions and honours

Professor Bracewell was a Fellow of the Royal Astronomical Society (1950), Fellow and life member of the Institute of Electrical and Electronic Engineers (1961), Fellow of the American Association for the Advancement of Science (1989), and was a Fellow with other significant societies and organisations.

For experimental contributions to the study of the ionosphere by means of very low frequency waves, Dr. Bracewell received the Duddell Premium of the Institution of Electrical Engineers, London in 1952. In 1992 he was elected to foreign associate membership of the Institute of Medicine of the U.S. National Academy of Sciences (1992), the first Australian to achieve that distinction, for fundamental contributions to medical imaging. He was one of Sydney University's three honourees when alumni awards were instituted in 1992, with a citation for brain scanning, and was the 1994 recipient of the Institute of Electrical and Electronic Engineers' Heinrich Hertz medal for pioneering work in antenna aperture synthesis and image reconstruction as applied to radio astronomy and to computer-assisted tomography. In 1998 Dr. Bracewell was named Officer of the Order of Australia (AO) for service to science in the fields of radio astronomy and image reconstruction.

At CSIRO Radiophysics Laboratory, work that in 1942-1945 was classified appeared in a dozen reports. Activities included design, construction, and demonstration of voice-modulation equipment for a 10 cm magnetron (July 1943), a microwave triode oscillator at 25 cm using cylindrical cavity resonators, equipment designed for microwave radar in field use (wavemeter, echo box, thermistor power meter, etc.) and microwave measurement technique. Experience with numerical computation of fields in cavities led, after the war, to a Master of Engineering degree (1948) and the definitive publication on step discontinuities in radial transmission lines (1954).

While at the Cavendish Laboratory, Cambridge (1946–1950) Bracewell worked on observation and theory of upper atmospheric ionisation, contributing to experimental technique (1948), explaining solar effects (1949), and distinguishing two layers below the E-layer (1952), work recognised by the Duddell Premium.

At Stanford Professor Bracewell constructed a microwave spectroheliograph (1961), a large and complex radio telescope which produced daily temperature maps of the sun reliably for eleven years, the duration of a solar cycle. The first radio telescope to give output automatically in printed form, and therefore capable of worldwide dissemination by teleprinter, its daily solar weather maps received acknowledgement from NASA for support of the first manned landing on the moon.

Many fundamental papers on restoration (1954–1962), interferometry (1958–1974) and reconstruction (1956–1961) appeared along with instrumental and observational papers. By 1961 the radio-interferometer calibration techniques developed for the spectroheliograph first allowed an antenna system, with 52" fan beam, to equal the angular resolution of the human eye in one observation. With this beam the components of Cygnus A, spaced 100", were put directly in evidence without the need for repeated observations with variable spacing aperture synthesis interferometry.

The nucleus of the extragalactic source Centaurus A was resolved into two separate components whose right ascensions were accurately determined with a 2.3-minute fan beam at 9.1 cm. Knowing that Centaurus A was composite, Bracewell used the 6.7-minute beam of the Parkes Observatory 64 m radiotelescope at 10 cm to determine the separate declinations of the components and in so doing was the first to observe strong polarisation in an extragalactic source (1962), a discovery of fundamental significance for the structure and role of astrophysical magnetic fields. Subsequent observations made at Parkes by other observers with a 14-minute and wider beams at 21 cm and longer wavelengths, though not resolving the components, were compatible with the λ2 dependence expected from Faraday rotation if magnetic fields were the polarising agent.

A second major radiotelescope (1971) employing advanced concepts to achieve an angular resolution of 18 seconds of arc was designed and built at Stanford and applied to both solar and galactic studies. The calibration techniques for this leading-edge resolution passed into general use in radio interferometry via the medium of alumni.

Upon the discovery of the cosmic background radiation:

  • a remarkable observational limit of 1.7 millikelvins, with considerable theoretical significance for cosmology, was set on the anisotropy in collaboration with Ph. D. student E.K. Conklin (1967), and was not improved on for many years
  • the correct theory of a relativistic observer in a blackbody enclosure (1968) was given in the first of several papers by various authors obtaining the same result
  • the absolute motion of the Sun at 308 km/s through the cosmic background radiation was measured by Conklin in 1969, some years before independent confirmation.

With the advent of the space age, Bracewell became interested in celestial mechanics, made observations of the radio emission from Sputnik 1, and supplied the press with accurate charts predicting the path of Soviet satellites, which were perfectly visible, if you knew when and where to look. Following the puzzling performance of Explorer I in orbit, he published the first explanation (1958-9) of the observed spin instability of satellites, in terms of the Poinsot motion of a non-rigid body with internal friction. He recorded the signals from Sputniks I, II and III and discussed them in terms of the satellite spin, antenna polarisation, and propagation effects of the ionised medium, especially Faraday effect.

Later (1978, 1979) he invented a spinning, nulling, two-element infrared interferometer suitable for space-shuttle launching into an orbit near Jupiter, with milliarcsecond resolution, that could lead to the discovery of planets around stars other than the sun. This concept was elaborated in 1995 by Angel and Woolf, whose space-station version with four-element double nulling became the Terrestrial Planet Finder (TPF), NASA's candidate for imaging planetary configurations of other stars.[1]

Imaging in astronomy led to participation in development of computer assisted x-ray tomography, where commercial scanners reconstruct tomographic images using the algorithm developed by Bracewell for radioastronomical reconstruction from fan-beam scans. This corpus of work has been recognized by the Institute of Medicine, an award by the University of Sydney, and the Heinrich Hertz medal. Service on the founding editorial board of the Journal for Computer-Assisted Tomography, to which he also contributed publications, and on the scientific advisory boards of medical instrumentation companies maintained Bracewell's interest in medical imaging, which became an important part of his regular graduate lectures on imaging, and forms an important part of his 1995 text on imaging.

Experience with the optics, mechanics and control of radiotelescopes led to involvement with solar thermophotovoltaic energy at the time of the energy crisis, including the fabrication of low-cost solid and perforated paraboloidal reflectors by hydraulic inflation.

Bracewell is also known for being the first to propose the use of autonomous interstellar space probes for communication between alien civilisations as an alternative to radio transmission dialogs. This hypothetical concept has been dubbed the Bracewell probe after its inventor.

Fourier analysis

As a consequence of relating images to Fourier analysis, in 1983 he discovered a new factorisation of the discrete Fourier transform matrix leading to a fast algorithm for spectral analysis. This method, which has advantages over the fast Fourier algorithm, especially for images, is treated in The Hartley Transform (1986), in U.S. Patent 4,646,256 (1987, now in the public domain), and in over 200 technical papers by various authors that were stimulated by the discovery. Analogue methods of creating a Hartley transform plane first with light and later with microwaves were demonstrated in the laboratory and permitted the determination of electromagnetic phase by the use of square-law detectors. A new elementary signal representation, the Chirplet transform, was discovered (1991) that complements the Gabor elementary signal representations used in dynamic spectral analysis (with the property of meeting the bandwidth-duration minimum associated with the uncertainty principle). This advance opened a new field of adaptive dynamic spectra with wide application in information analysis.

Other interests

Professor Bracewell was interested in conveying an appreciation of the role of science in society to the public, in mitigating the effects of scientific illiteracy on public decision making through contact with alumni groups, and in liberal undergraduate education within the framework of the Astronomy Course Program and the Western Culture program in Values, Technology, Science and Society, in both of which he taught for some years. He gave the 1996 Bunyan Lecture on The Destiny of Man.

He was also interested in the trees of Stanford's campus and published a book about them. He also taught an undergraduate seminar titled I Dig Trees.[2][3]

Bracewell was also a designer and builder of sundials. He built one on the South side of the Terman Engineering Building. He built one at the home of his son, Mark Bracewell. He built another on the deck of professor John Linvill's house.

As his seminar "I Dig Trees" indicated, Dr. Bracewell was known for having a tremendously keen, intelligent sense of wry, science-infused humor. One of his treasured family photos showed him sitting on the ground, legs akimbo, with a beer bottle in front of him that he had neatly balanced on one of its bottom edges—his proof that even that thin edge had 3 balance points.

Selected publications

  • Bracewell, R.N. and Pawsey, J.L., Radio Astronomy (Oxford, 1955) (also translated into Russian and reprinted in China)
  • Bracewell, R.N., Radio Interferometry of Discrete Sources (Proceedings of the IRE, January 1958)
  • Bracewell, Ronald N., ed., Paris Symposium on Radio Astronomy, IAU Symposium no. 9 and URSI Symposium no. 1, held 30 July 1958 – 6 August 1958 (Stanford Univ. Press, Stanford, CA, 1959) (also translated into Russian)
  • Professor Bracewell translated Radio Astronomy, by J.L. Steinberg and J. Lequeux, (McGraw-Hill, 1963) from French
  • Bracewell, R.N., The Fourier Transform and Its Applications (McGraw-Hill, 1965, 2nd ed. 1978, revised 1986) (also translated into Japanese and Polish)
  • Bracewell, R.N., Trees on the Stanford Campus (Stanford: Samizdat, 1973)
  • Bracewell, R.N., The Galactic Club: Intelligent Life in Outer Space (Portable Stanford: Alumni Association, 1974) (also translated into Dutch, Japanese, and Italian)
  • Bracewell, R.N., The Hartley Transform (Oxford University Press, 1986) (also translated into German and Russian)
  • Bracewell, R.N., Two-Dimensional Imaging (Prentice-Hall, 1995)
  • Bracewell, R.N., Fourier Analysis and Imaging (Plenum, 2004)
  • Bracewell, R.N., Trees of Stanford and Environs (Stanford Historical Society, 2005)

Chapter contributions

Bracewell has contributed chapters to:

  • Textbook of Radar Microwave Transmission and Cavity Resonator Theory, ed. E.G. Bowen, 1946
  • Advances in Astronautical Sciences Satellite Rotation, ed. H. Jacobs, 1959
  • The Radio Noise Spectrum Correcting Noise Maps for Beamwidth, ed. D.H. Menzel, 1960
  • Modern Physics for the Engineer Radio Astronomy, ed. L. Ridenour and W. Nierenberg, 1960
  • Statistical methods in Radio Wave Propagation Antenna Tolerance Theory, ed. W.C. Hoffman, 1960
  • Advances in Geophysics Satellite Studies of the Ionization in Space by Radio, ed. H.E. Landsberg, 1961 (O.K. Garriott and R.N. Bracewell)
  • Handbuch der Physik Radio Astronomy Techniques, ed. S. Flugge, 1962
  • Encyclopedia of Electronics Extraterrestrial Radio Noise, ed. C. Susskind, 1962
  • Stars and Galaxies Radio Broadcasts from the Depths of Space, ed. T.L Page, 1962
  • Radio Waves and Circuits Aerials and Data Processing, ed. S. Silver, 1963
  • Light and Life in the Universe Life in the Galaxy, ed. S.T. Butler and H. Messel, 1964
  • Encyclopædia Britannica Telescope, Radio, 1967
  • Vistas in Science The Microwave Sky, ed. David L. Arm, 1968
  • Man in Inner and Outer Space The Sun (Five Chapters), ed. H. Messel and S.T. Butler, 1968
  • Image Reconstruction from Projections: Implementation and Applications Image Reconstruction in Radio Astronomy, ed. G. Hermann, 1979
  • Annual Review of Astronomy and Astrophysics Computer Image Processing, ed. G. Burbidge et al., 1979
  • Energy for Survival How It All Began, Man the Lazy Animal, and Energy from Sunlight, ed. H. Messel, 1979
  • Life in the Universe Manifestations of Advanced Civilizations, ed. J. Billingham, 1981
  • Extraterrestrials: Where Are They? Preemption of the Galaxy by the First Advanced Civilization, ed. M.H. Hart and B. Zuckerman, 1982, 1995
  • Transformations in Optical Signal Processing Fourier Inversion of Deficient Data, ed. W.T. Rhodes, J.R. Fienup and B.E.A. Saleh, 1984
  • The Early Years of Radio Astronomy Early Work on Imaging Theory in Radio Astronomy, ed. W.T. Sullivan, III, 1984
  • Indirect Imaging Inversion of Nonplanar Visibilities, ed. J.A. Roberts, 1984
  • Fourier Techniques and Applications The Life of Joseph Fourier and Fourier Techniques and Applications, ed. J.F. Price, 1985
  • Yearbook of Science and Technology Wavelets, 1996
  • Encyclopedia of Applied Physics Fourier and Other Mathematical Transforms 1997
  • Cornelius Lanczos—Collected Published Papers with Commentaries The Fast Fourier Transform andSmoothing Data by Analysis and by Eye ed. W.R. Davis et al., 1999

References

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External links

Template:Persondata

fr:Ronald Bracewell

  1. ↑ Scientific American, April 1996
  2. ↑ Template:Cite web
  3. ↑ Template:Cite web
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