# Airy function

The Airy function describes a special function in mathematics. The function and its related function , also called the Airy function, are solutions to the linear differential equation${\ displaystyle \ operatorname {Ai} (x)}$${\ displaystyle \ operatorname {Ai} (x)}$${\ displaystyle \ operatorname {Bi} (x)}$

${\ displaystyle \ y '' - xy = 0 \,}$

also known as the Airy equation. Among other things, it describes the solution of the Schrödinger equation for a linear potential well .

The Airy function is named after the British astronomer George Biddell Airy , who used this function in his work in optics (Airy 1838). The term was introduced by Harold Jeffreys . ${\ displaystyle \ operatorname {Ai} (x)}$

## definition

For real values , the Airy function is defined as a parameter integral: ${\ displaystyle x}$

${\ displaystyle \ mathrm {Ai} (x) = {\ frac {1} {\ pi}} \ int \ limits _ {0} ^ {\ infty} \ cos \ left ({\ frac {t ^ {3} } {3}} + xt \ right) \, {\ rm {d}} t \.}$

A second, linearly independent solution of the differential equation is the Airy function of the second kind : ${\ displaystyle \ mathrm {Bi}}$

${\ displaystyle \ mathrm {Bi} (x) = {\ frac {1} {\ pi}} \ int \ limits _ {0} ^ {\ infty} \ left (\ exp \ left (- {\ frac {t ^ {3}} {3}} + xt \ right) + \ sin \ left ({\ frac {t ^ {3}} {3}} + xt \ right) \ right) \, {\ rm {d} } t \.}$

## properties

### Asymptotic behavior

For to be and using the WKB approximation approximate: ${\ displaystyle x}$${\ displaystyle + \ infty}$${\ displaystyle \ mathrm {Ai} (x)}$${\ displaystyle \ mathrm {Bi} (x)}$

{\ displaystyle {\ begin {aligned} \ mathrm {Ai} (x) & {} \ simeq {\ frac {e ^ {- {\ frac {2} {3}} x ^ {3/2}}} { 2 {\ sqrt {\ pi}} \, x ^ {1/4}}} \\\ mathrm {Bi} (x) & {} \ simeq {\ frac {e ^ {{\ frac {2} {3 }} x ^ {3/2}}} {{\ sqrt {\ pi}} \, x ^ {1/4}}}. \ end {aligned}}}

The following relationships apply to against : ${\ displaystyle x}$${\ displaystyle - \ infty}$

{\ displaystyle {\ begin {aligned} \ mathrm {Ai} (x) & {} \ simeq {\ frac {\ sin ({\ frac {2} {3}} (- x) ^ {3/2} + {\ frac {1} {4}} \ pi)} {{\ sqrt {\ pi}} \, (- x) ^ {1/4}}} \\\ mathrm {Bi} (x) & {} \ simeq {\ frac {\ cos ({\ frac {2} {3}} (- x) ^ {3/2} + {\ frac {1} {4}} \ pi)} {{\ sqrt {\ pi}} \, (- x) ^ {1/4}}}. \ end {aligned}}}

### zeropoint

The Airy functions only have zeros on the negative real axis. The approximate position follows from the asymptotic behavior for to ${\ displaystyle x \ to - \ infty}$

${\ displaystyle \ operatorname {Ai} (x) = 0 \ quad \ Rightarrow \ quad x \ approx - {\ bigl (} \ textstyle {\ frac {3} {2}} \ pi (n - {\ frac {1 } {4}}) {\ bigr)} ^ {2/3}, \ quad n \ in \ mathbb {N}}$
${\ displaystyle \ operatorname {Bi} (x) = 0 \ quad \ Rightarrow \ quad x \ approx - {\ bigl (} \ textstyle {\ frac {3} {2}} \ pi (n - {\ frac {3 } {4}}) {\ bigr)} ^ {2/3}, \ quad n \ in \ mathbb {N}}$

### Special values

The Airy functions and their derivatives have the following values: ${\ displaystyle x = 0}$

{\ displaystyle {\ begin {aligned} \ mathrm {Ai} (0) & {} = {\ frac {1} {{\ sqrt [{3}] {9}} \ cdot \ Gamma ({\ frac {2 } {3}})}}, & \ quad \ mathrm {Ai} '(0) & {} = - {\ frac {1} {{\ sqrt [{3}] {3}} \ cdot \ Gamma ( {\ frac {1} {3}})}}, \\\ mathrm {Bi} (0) & {} = {\ frac {1} {{\ sqrt [{6}] {3}} \ cdot \ Gamma ({\ frac {2} {3}})}}, & \ quad \ mathrm {Bi} '(0) & {} = {\ frac {\ sqrt [{6}] {3}} {\ Gamma ({\ frac {1} {3}})}}. \ end {aligned}}}

This is the gamma function . It follows that the Wronsky determinant of and is equal . ${\ displaystyle \ Gamma (\ cdot)}$${\ displaystyle \ mathrm {Ai} (x)}$${\ displaystyle \ mathrm {Bi} (x)}$${\ displaystyle {\ tfrac {1} {\ pi}}}$

## Fourier transform

Its Fourier transform follows directly from the definition of the Airy function (see above) . ${\ displaystyle \ operatorname {Ai} (x)}$

${\ displaystyle {\ mathcal {F}} (\ operatorname {Ai}) (k): = \ int _ {- \ infty} ^ {\ infty} \ operatorname {Ai} (x) \ \ mathrm {e} ^ {-2 \ pi \ mathrm {i} kx} \, dx = \ mathrm {e} ^ {{\ frac {\ mathrm {i}} {3}} (2 \ pi k) ^ {3}} \, .}$

Note the symmetrical variant of the Fourier transform used here.

## Further representations

${\ displaystyle \ mathrm {Ai} (z) = {\ frac {1} {3 ^ {2/3} \ cdot \ Gamma ({\ tfrac {2} {3}})}} \ cdot \, {} _ {0} F_ {1} \ left (0; {\ tfrac {2} {3}}; {\ tfrac {1} {9}} z ^ {3} \ right) - {\ frac {z} { 3 ^ {1/3} \ cdot \ Gamma ({\ tfrac {1} {3}})}} \ cdot \, {} _ {0} F_ {1} \ left (0; {\ tfrac {4} {3}}; {\ tfrac {1} {9}} z ^ {3} \ right)}$
${\ displaystyle \ mathrm {Bi} (z) = {\ frac {1} {3 ^ {1/6} \ cdot \ Gamma ({\ tfrac {2} {3}})}} \ cdot \, {} _ {0} F_ {1} \ left (0; {\ tfrac {2} {3}}; {\ tfrac {1} {9}} z ^ {3} \ right) + {\ frac {3 ^ { 1/6} \ cdot z} {\ Gamma ({\ tfrac {1} {3}})}} \ cdot \, {} _ {0} F_ {1} \ left (0; {\ tfrac {4} {3}}; {\ tfrac {1} {9}} z ^ {3} \ right)}$
${\ displaystyle \ mathrm {Ai} (x) = {\ frac {1} {3}} {\ sqrt {x}} \ left [I _ {- 1/3} \ left ({\ frac {2} {3 }} x ^ {3/2} \ right) -I_ {1/3} \ left ({\ frac {2} {3}} x ^ {3/2} \ right) \ right]}$
${\ displaystyle \ mathrm {Bi} (x) = {\ sqrt {\ frac {x} {3}}} \ left [I _ {- 1/3} \ left ({\ frac {2} {3}} x ^ {3/2} \ right) + I_ {1/3} \ left ({\ frac {2} {3}} x ^ {3/2} \ right) \ right]}$
• Another infinite integral representation for is${\ displaystyle \ mathrm {Ai}}$
${\ displaystyle \ mathrm {Ai} (z) = {\ frac {1} {2 \ pi}} \ int \ limits _ {- \ infty} ^ {\ infty} \ exp \ left (\ mathrm {i} \ cdot \ left (zt + {\ frac {t ^ {3}} {3}} \ right) \ right) \ mathrm {d} t}$
• There are the series representations
${\ displaystyle \ mathrm {Ai} (z) = {\ frac {1} {3 ^ {2/3} \ pi}} \ sum _ {n = 0} ^ {\ infty} {\ frac {\ Gamma \ left ({\ frac {1} {3}} (n + 1) \ right)} {n!}} \ left (3 ^ {1/3} z \ right) ^ {n} \ sin \ left ({ \ frac {2 (n + 1) \ pi} {3}} \ right)}$
${\ displaystyle \ mathrm {Bi} (z) = {\ frac {1} {3 ^ {1/6} \ pi}} \ sum _ {n = 0} ^ {\ infty} {\ frac {\ Gamma \ left ({\ frac {1} {3}} (n + 1) \ right)} {n!}} \ left (3 ^ {1/3} z \ right) ^ {n} \ left | \ sin \ left ({\ frac {2 (n + 1) \ pi} {3}} \ right) \ right |}$

## Complex arguments

${\ displaystyle \ mathrm {Ai} (x)}$and are whole functions . So they can be continued analytically on the entire complex level. ${\ displaystyle \ mathrm {Bi} (x)}$

${\ displaystyle \ Re \ left [\ mathrm {Ai} (x + iy) \ right]}$ ${\ displaystyle \ Im \ left [\ mathrm {Ai} (x + iy) \ right]}$ ${\ displaystyle | \ mathrm {Ai} (x + iy) | \,}$ ${\ displaystyle \ mathrm {arg} \ left [\ mathrm {Ai} (x + iy) \ right] \,}$

${\ displaystyle \ Re \ left [\ mathrm {Bi} (x + iy) \ right]}$ ${\ displaystyle \ Im \ left [\ mathrm {Bi} (x + iy) \ right]}$ ${\ displaystyle | \ mathrm {Bi} (x + iy) | \,}$ ${\ displaystyle \ mathrm {arg} \ left [\ mathrm {Bi} (x + iy) \ right] \,}$

## Related functions

### Airy zeta function

Analogous to the other zeta functions, the Airy zeta function can be defined for the Airy function as

${\ displaystyle Z (n) = \ sum _ {r} {\ frac {1} {r ^ {n}}},}$

where the sum goes over the real (negative) zeros of . ${\ displaystyle \ mathrm {Ai}}$

### Scorer functions

Function graphs of and .${\ displaystyle \ mathrm {Gi} (x)}$${\ displaystyle \ mathrm {Hi} (x)}$

Sometimes the two other functions and are added to the Airy functions. The integral definitions are ${\ displaystyle \ mathrm {Gi} (x)}$${\ displaystyle \ mathrm {Hi} (x)}$

${\ displaystyle \ mathrm {Gi} (x) = {\ frac {1} {\ pi}} \ int \ limits _ {0} ^ {\ infty} \ sin \ left ({\ frac {t ^ {3} } {3}} + xt \ right) \, \ mathrm {d} t}$
${\ displaystyle \ mathrm {Hi} (x) = {\ frac {1} {\ pi}} \ int \ limits _ {0} ^ {\ infty} \ exp \ left (- {\ frac {t ^ {3 }} {3}} + xt \ right) \, \ mathrm {d} t}$

They can also be represented by the functions and . ${\ displaystyle \ mathrm {Ai}}$${\ displaystyle \ mathrm {Bi}}$