Application To The Schwarzian Equation
An important application of the Riccati equation is to the 3rd order Schwarzian differential equation
which occurs in the theory of conformal mapping and univalent functions. In this case the ODEs are in the complex domain and differentiation is with respect to a complex variable. (The Schwarzian derivative has the remarkable property that it is invariant under Möbius transformations, i.e. whenever is non-zero.) The function satisfies the Riccati equation
By the above where is a solution of the linear ODE
Since, integration gives for some constant . On the other hand any other independent solution of the linear ODE has constant non-zero Wronskian which can be taken to be after scaling. Thus
so that the Schwarzian equation has solution
Read more about this topic: Riccati Equation
Famous quotes containing the words application to, application and/or equation:
““Five o’clock tea” is a phrase our “rude forefathers,” even of the last generation, would scarcely have understood, so completely is it a thing of to-day; and yet, so rapid is the March of the Mind, it has already risen into a national institution, and rivals, in its universal application to all ranks and ages, and as a specific for “all the ills that flesh is heir to,” the glorious Magna Charta.”
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“If you would be a favourite of your king, address yourself to his weaknesses. An application to his reason will seldom prove very successful.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“A nation fights well in proportion to the amount of men and materials it has. And the other equation is that the individual soldier in that army is a more effective soldier the poorer his standard of living has been in the past.”
—Norman Mailer (b. 1923)