We apply the uniform asymptotics method to the fifth Painlev&#233; transcendants, find its asymptotics of the form <mml:math alttext="$y=-1+t^{-1/2}A(t)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mo>&#8722;</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mrow><mml:mo>&#8722;</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> as <mml:math alttext="$t\rightarrow\infty$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#8594;</mml:mo><mml:mi>&#8734;</mml:mi></mml:mrow></mml:math> along the positive <mml:math alttext="$t$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>t</mml:mi></mml:math>-axis, and obtain the corresponding monodromy data.

International Journal of Mathematics and Mathematical Sciences

Application of uniform asymptotics to the fifth Painlevé transcendant

Abstract

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