Stokes lines
E620766
concept in asymptotic analysis
concept in complex analysis
concept in differential equations
mathematical concept
Stokes lines are specific curves in the complex plane across which the asymptotic behavior of solutions to differential equations changes, playing a key role in asymptotic analysis and complex function theory.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
concept in asymptotic analysis
ⓘ
concept in complex analysis ⓘ concept in differential equations ⓘ mathematical concept ⓘ |
| appearsIn |
saddle point analysis of contour integrals
ⓘ
theory of linear ODEs with irregular singular points ⓘ uniform asymptotic approximations near turning points ⓘ |
| context |
complex plane of the independent variable
ⓘ
complex plane of the phase function in integral representations ⓘ |
| definition |
curves in the complex plane across which the dominant asymptotic behavior of a solution changes
ⓘ
loci in the complex plane where exponentially small contributions in an asymptotic expansion switch on or off ⓘ |
| distinguishedFrom | anti-Stokes lines, where the real part of the exponent is constant ⓘ |
| effect |
cause discontinuous changes in asymptotic coefficients across them
ⓘ
control the activation of exponentially small terms in asymptotic series ⓘ |
| field |
asymptotic analysis
ⓘ
complex analysis ⓘ ordinary differential equations ⓘ special functions ⓘ |
| mathematicalFormulation |
for exponentials e^{
phi(z)/
ε}, Stokes lines satisfy Im(φ(z)) = constant where Re(φ(z)) changes sign
ⓘ
often defined by conditions on the argument of a complex phase function ⓘ |
| namedAfter | George Gabriel Stokes NERFINISHED ⓘ |
| property |
are defined relative to a particular asymptotic expansion or solution
ⓘ
are typically curves along which the imaginary part of an exponent is constant ⓘ depend on the choice of branch of multivalued functions ⓘ mark where exponentially subdominant terms become comparable to dominant terms ⓘ often emanate from singularities or turning points of the differential equation ⓘ |
| relatedTo |
Stokes phenomenon
NERFINISHED
ⓘ
WKB approximation ⓘ analytic continuation ⓘ anti-Stokes lines ⓘ asymptotic expansion ⓘ connection formulas ⓘ saddle point method ⓘ singular points of differential equations ⓘ steepest descent paths ⓘ turning points of differential equations ⓘ |
| role |
determine sectors of validity of asymptotic expansions
ⓘ
govern the switching of asymptotic contributions in Stokes phenomenon ⓘ guide analytic continuation of asymptotic solutions ⓘ separate regions with different asymptotic dominance of terms ⓘ |
| usedIn |
WKB analysis of Schrödinger-type equations
ⓘ
analysis of Bessel functions ⓘ analysis of special functions such as Airy functions ⓘ asymptotic analysis of linear differential equations ⓘ exponential asymptotics ⓘ matched asymptotic expansions ⓘ resurgent analysis ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.