`:top

In mathematics, the `!Bateman function`! (or `*k`*-function) is a special case of the `F33f`_`[confluent hypergeometric function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Confluent_hypergeometric_function]`_`f studied by `F33f`_`[Harry Bateman`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Harry_Bateman]`_`f(1931).`:cite-ref-1[`F5bf`_`[1`#cite-note-1]`_`f]`:cite-ref-2[`F5bf`_`[2`#cite-note-2]`_`f] Bateman defined it by

k ν ν ( x ) = 2 π π ∫ ∫ 0 π π / 2 cos ⁡ ⁡ ( x tan ⁡ ⁡ θ θ − − ν ν θ θ ) d θ θ . {\\displaystyle \\displaystyle k_{\\nu }(x)={\\frac {2}{\\pi }}\\int _{0}^{\\pi /2}\\cos(x\\tan \\theta -\\nu \\theta )\\,d\\theta .}

`F33f`_`[Bateman`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Harry_Bateman]`_`f discovered this function, when `F33f`_`[Theodore von Kármán`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Theodore_von_Kármán]`_`f asked for the solution of the following differential equation which appeared in the theory of `F33f`_`[turbulence`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Turbulence]`_`f`:cite-ref-3[`F5bf`_`[3`#cite-note-3]`_`f]

x d 2 u d x 2 = ( x − − ν ν ) u {\\displaystyle x{\\frac {d^{2}u}{dx^{2}}}=(x-\\nu )u}

and Bateman found this function as one of the solutions. Bateman denoted this function as "k" function in honor of `F33f`_`[Theodore von Kármán`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Theodore_von_Kármán]`_`f.

The Bateman function for x > 0 {\\displaystyle x>0} is the related to the `F33f`_`[Confluent hypergeometric function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Confluent_hypergeometric_function]`_`f of the second kind as follows

k ν ν ( x ) = e − − x Γ Γ ( 1 + 1 2 ν ν ) U ( − − 1 2 ν ν , 0 , 2 x ) , x > 0. {\\displaystyle k_{\\nu }(x)={\\frac {e^{-x}}{\\Gamma \\left(1+{\\frac {1}{2}}\\nu \\right)}}U\\left(-{\\frac {1}{2}}\\nu ,0,2x\\right),\\quad x>0.}

This is not to be confused with another function of the same name which is used in Pharmacokinetics.

>>Contents

• `F0af`_`[Havelock function`#havelock-function]`_`f

• `F0af`_`[Properties`#properties]`_`f

• `F0af`_`[References`#references]`_`f

-─

>>Havelock function

Complementary to the Bateman function, one may also define the Havelock function, named after `F33f`_`[Thomas Henry Havelock`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Thomas_Henry_Havelock]`_`f. In fact, both the Bateman and the Havelock functions were first introduced by Havelock in 1927,`:cite-ref-4[`F5bf`_`[4`#cite-note-4]`_`f] while investigating the surface elevation of the uniform stream past an immersed circular cylinder. The Havelock function is defined by

h ν ν ( x ) = 2 π π ∫ ∫ 0 π π / 2 sin ⁡ ⁡ ( x tan ⁡ ⁡ θ θ − − ν ν θ θ ) d θ θ . {\\displaystyle \\displaystyle h_{\\nu }(x)={\\frac {2}{\\pi }}\\int _{0}^{\\pi /2}\\sin(x\\tan \\theta -\\nu \\theta )\\,d\\theta .}

>>Properties

• k 0 ( x ) = e − − | x | {\\displaystyle k_{0}(x)=e^{-|x|}}

• k − − n ( x ) = k n ( − − x ) {\\displaystyle k_{-n}(x)=k_{n}(-x)}

• k n ( 0 ) = 2 n π π sin ⁡ ⁡ n π π 2 {\\displaystyle k_{n}(0)={\\frac {2}{n\\pi }}\\sin {\\frac {n\\pi }{2}}}

• k 2 ( x ) = ( x + | x | ) e − − | x | {\\displaystyle k_{2}(x)=(x+|x|)e^{-|x|}}

• | k n ( x ) | ≤ ≤ 1 {\\displaystyle |k_{n}(x)|\\leq 1} for real values of n {\\displaystyle n} and x {\\displaystyle x}

• k 2 n ( x ) = 0 {\\displaystyle k_{2n}(x)=0} for x < 0 {\\displaystyle x<0} if n {\\displaystyle n} is a positive integer

• k 1 ( x ) = 2 x π π [ K 1 ( x ) + K 0 ( x ) ] , x < 0 {\\displaystyle k_{1}(x)={\\frac {2x}{\\pi }}[K_{1}(x)+K_{0}(x)],\\ x<0} , where K n ( − − x ) {\\displaystyle K_{n}(-x)} is the `F33f`_`[Modified Bessel function of the second kind`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Modified_Bessel_function_of_the_second_kind]`_`f

>>References

`:cite-note-1`!1.`! `F0af`_`[↑`#cite-ref-1]`_`f `:citerefbateman1931`a`F33f`_`[Bateman, H.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Harry_Bateman]`_`f (1931), "The k-function, a particular case of the confluent hypergeometric function", `*`F33f`_`[Transactions of the American Mathematical Society`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Transactions_of_the_American_Mathematical_Society]`_`f`*, `!33`! (4): 817–831, `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.2307/1989510, `F33f`_`[ISSN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISSN_(identifier)]`_`f 0002-9947, `F33f`_`[JSTOR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=JSTOR_(identifier)]`_`f 1989510, `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 1501618

`:cite-note-2`!2.`! `F0af`_`[↑`#cite-ref-2]`_`f "Bateman function", `*`F33f`_`[Encyclopedia of Mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Encyclopedia_of_Mathematics]`_`f`*, `F33f`_`[EMS Press`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=European_Mathematical_Society]`_`f, 2001 [1994]

`:cite-note-3`!3.`! `F0af`_`[↑`#cite-ref-3]`_`f Martin, P. A., & Bateman, H. (2010). from Manchester to Manuscript Project. Mathematics Today, 46, 82-85. http://www.math.ust.hk/~machiang/papers_folder/http___www.ima.org.uk_mathematics_mt_april10_harry_bateman_from_manchester_to_manuscript_project.pdf

`:cite-note-4`!4.`! `F0af`_`[↑`#cite-ref-4]`_`f Havelock, T. H. (1927). The method of images in some problems of surface waves. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 115(771), 268-280.

`c`F0af`_`[↑ Back to top`#top]`_`f`a