The angle property of positive real functions simply derived

Helge Jørsboe

The angle property of positive real functions simply derived

Helge Jørsboe

Technical University of Denmark

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Abstract

The angle property of positive real (rational) functionsZ(s), namely, that|arg s | geqq |arg Z(s)|in the right half of thes-plane, can be demonstrated very simply by an examination of the imaginary parts of the functionsln(s/Z(s))andln (sZ(s)), i.e.,arg s mp arg Z(s). In particular, on a contour enclosing the entire first quadrant,arg s mp arg Z(s)can rather easily be shown to be nonnegative. The extremum theorem of analytic functions then assures thatarg s mp arg Z(s)cannot be negative inside the first quadrant; thus the angle property is demonstrated in the first quadrant. The same result is obtained immediately in the fourth quadrant.

Original language	English
Journal	IEEE Transactions on Circuits and Systems
Volume	20
Issue number	3
Pages (from-to)	327-328
ISSN	0098-4094
Publication status	Published - 1973

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Copyright: 1973 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE

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title = "The angle property of positive real functions simply derived",

abstract = "The angle property of positive real (rational) functionsZ(s), namely, that|arg s | geqq |arg Z(s)|in the right half of thes-plane, can be demonstrated very simply by an examination of the imaginary parts of the functionsln(s/Z(s))andln (sZ(s)), i.e.,arg s mp arg Z(s). In particular, on a contour enclosing the entire first quadrant,arg s mp arg Z(s)can rather easily be shown to be nonnegative. The extremum theorem of analytic functions then assures thatarg s mp arg Z(s)cannot be negative inside the first quadrant; thus the angle property is demonstrated in the first quadrant. The same result is obtained immediately in the fourth quadrant.",

author = "Helge J{\o}rsboe",

note = "Copyright: 1973 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE",

year = "1973",

language = "English",

volume = "20",

pages = "327--328",

journal = "IEEE Transactions on Circuits and Systems",

issn = "0098-4094",

publisher = "Institute of Electrical and Electronics Engineers Inc.",

number = "3",

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TY - JOUR

T1 - The angle property of positive real functions simply derived

AU - Jørsboe, Helge

N1 - Copyright: 1973 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE

PY - 1973

Y1 - 1973

N2 - The angle property of positive real (rational) functionsZ(s), namely, that|arg s | geqq |arg Z(s)|in the right half of thes-plane, can be demonstrated very simply by an examination of the imaginary parts of the functionsln(s/Z(s))andln (sZ(s)), i.e.,arg s mp arg Z(s). In particular, on a contour enclosing the entire first quadrant,arg s mp arg Z(s)can rather easily be shown to be nonnegative. The extremum theorem of analytic functions then assures thatarg s mp arg Z(s)cannot be negative inside the first quadrant; thus the angle property is demonstrated in the first quadrant. The same result is obtained immediately in the fourth quadrant.

AB - The angle property of positive real (rational) functionsZ(s), namely, that|arg s | geqq |arg Z(s)|in the right half of thes-plane, can be demonstrated very simply by an examination of the imaginary parts of the functionsln(s/Z(s))andln (sZ(s)), i.e.,arg s mp arg Z(s). In particular, on a contour enclosing the entire first quadrant,arg s mp arg Z(s)can rather easily be shown to be nonnegative. The extremum theorem of analytic functions then assures thatarg s mp arg Z(s)cannot be negative inside the first quadrant; thus the angle property is demonstrated in the first quadrant. The same result is obtained immediately in the fourth quadrant.

M3 - Letter

SN - 0098-4094

VL - 20

SP - 327

EP - 328

JO - IEEE Transactions on Circuits and Systems

JF - IEEE Transactions on Circuits and Systems

IS - 3

ER -

The angle property of positive real functions simply derived

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