scholarly journals Equi-Distribution of Values for the Third and the Fifth Painlevé Transcendents

2008 ◽  
Vol 192 ◽  
pp. 89-109 ◽  
Author(s):  
Shun Shimomura
Keyword(s):  
Characteristic Function ◽  
Growth Order ◽  
Value Distribution Theory ◽  
Painlevé Transcendents ◽  
The Third ◽  
Domain Type ◽  
Distribution Of Values ◽  
Special Cases ◽  
Painleve Transcendents ◽  
Borel Exceptional Values

AbstractWe show equi-distribution properties of values for the third and the fifth Painlevé transcendents in a sectorial domain. For our purpose we define a characteristic function of sectorial domain type by employing value distribution theory in a half plane. Some special cases admit analogues of Borel exceptional values. Similar results are obtained for modified versions of these Painlevé transcendents, which are of infinite growth order.

The meromorphic nature of the sixth Painlevé transcendents

2004 ◽  
Vol 94 (1) ◽  
pp. 319-342 ◽  
Author(s):  
Aimo Hinkkanen ◽  
Ilpo Laine
Keyword(s):  
Painlevé Transcendents ◽  
Painleve Transcendents

Aberrations of Volume Cylindrical Holographic Optical Element

2013 ◽  
Vol 8 (3) ◽  
pp. 6-12
Author(s):  
Yuriy Batomunkuev
Keyword(s):  
Refractive Index ◽  
Characteristic Function ◽  
Optical Element ◽  
Thermally Induced ◽  
The Third ◽  
Holographic Optical Element ◽  
Seventh Order ◽  
Chromatic Aberrations ◽  
Cylindrical Volume ◽  
Analytical Expressions

The analytical expressions allowed to calculate the third-, fifth- and seventh-order monochromatic and chromatic aberrations are obtained for the cylindrical volume holographic optical element by method of the characteristic function. The formulas for coefficients of third-, fifth- and seventh-order aberrations are presented. It is noted that coefficients of the aberrations arising because of photo induced, thermally induced and deformation changes of refractive index and of sizes of the cylindrical volume holographic optical element can be isolated in these coefficients. It is shown that width of the working spectral range for reflection cylindrical volume holographic optical element is inversely proportional to its thickness and for transmission holographic element is inversely proportional to square its thickness

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