# Question about MarcumQ function **URL:** \ **Category:** Questions\ **Tags:** mathematica, statistics-and-probability\ **Created:** [December 24, 2014, 6:59pm UTC](https://community.wolfram.com/t/question-about-marcumq-function/4388 "2014-12-24T18:59:29Z")\ **Posts on this page:** 9\ **Page:** 1
**Author:** ![safnhd](https://avatars.discourse-cdn.com/v4/letter/s/3ab097/32.png) [@safnhd](https://community.wolfram.com/u/safnhd)\ **Post date:** [December 24, 2014, 6:59pm UTC](https://community.wolfram.com/t/question-about-marcumq-function/4388/1 "2014-12-24T18:59:29Z")
Hi, I am trying to solve an equation that has a MarcumQ function in it. One of the MarcumQ properties is that ``` Limit[MarcumQ[1, 3, \[Mu]], \[Mu] -> Infinity] = 1 ``` This is obvious when the function is plotted while varying its arguments ``` Plot[{MarcumQ[1, 1, \[Mu]], MarcumQ[1, 2, \[Mu]], MarcumQ[1, 3, \[Mu]], MarcumQ[1, 4, \[Mu]], MarcumQ[1, 5, \[Mu]]}, {\[Mu], 0, 8}] ``` However, the Mathematica Limit function seems to be unable to give an answer. I wonder, what the reason could be. Thanks in advance. Saf ---
**Author:** ![dreiss](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/dreiss/32/29094_2.png) [@dreiss](https://community.wolfram.com/u/dreiss)\ **Post date:** [December 25, 2014, 3:51am UTC](https://community.wolfram.com/t/question-about-marcumq-function/4388/2 "2014-12-25T03:51:14Z")
It appears that Mathematica does not know this information about MarcumQ. It also seems not to be able to compute a power series expansion about `mu=0`, oddly just returning the MarcumQ function it self: ``` In[1]:= Series[MarcumQ[1, 3, \[Mu]], {\[Mu], 0, 2}] Out[1]= MarcumQ[1, 3, \[Mu]] ``` This is so, even though Mathematica knows how to compute derivatives, e.g.: ``` In[2]:= D[MarcumQ[1, 3, \[Mu]], {\[Mu], 3}] Out[2]= 3 E^(1/2 (-9 - \[Mu]^2)) \[Mu] Hypergeometric0F1Regularized[1, ( 9 \[Mu]^2)/4] - E^(1/2 (-9 - \[Mu]^2)) \[Mu]^3 Hypergeometric0F1Regularized[1, ( 9 \[Mu]^2)/4] - 27/2 E^(1/2 (-9 - \[Mu]^2)) \[Mu] Hypergeometric0F1Regularized[2, ( 9 \[Mu]^2)/4] + 9 E^(1/2 (-9 - \[Mu]^2)) \[Mu]^3 Hypergeometric0F1Regularized[2, ( 9 \[Mu]^2)/4] - 81/4 E^(1/2 (-9 - \[Mu]^2)) \[Mu]^3 Hypergeometric0F1Regularized[3, ( 9 \[Mu]^2)/4] ``` ---
**Author:** ![sukrause](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/sukrause/32/41348_2.png) [@sukrause](https://community.wolfram.com/u/sukrause)\ **Post date:** [December 25, 2014, 10:19am UTC](https://community.wolfram.com/t/question-about-marcumq-function/4388/3 "2014-12-25T10:19:41Z")
0 is not positive, the manual states ![comm411586.PNG](https://global.discourse-cdn.com/wolfram/original/3X/0/3/03952496f9f2f06c20d5ec2d0fe5b7da04b9c06f.png) next thing the limit b\rightarrow\infty (or \mu\rightarrow\infty) requests the integral \int\_\infty^\infty ... dx why this should evaluate to 1? ---
**Author:** ![safnhd](https://avatars.discourse-cdn.com/v4/letter/s/3ab097/32.png) [@safnhd](https://community.wolfram.com/u/safnhd)\ **Post date:** [December 26, 2014, 5:10am UTC](https://community.wolfram.com/t/question-about-marcumq-function/4388/4 "2014-12-26T05:10:53Z")
Hi, you are right about the zero in the plotted range. I should have started the plot from a real number, which is a little bit bigger than zero. It seems that didn’t make any difference though. However, as seen in the example, I included in the my question, the value of the marcumQ function evaluates, indeed, to 1, when the value of the second argument goes to infinity. The book I am reading for the time being, confirms that as well. But, you have a point in presenting the mathematical definition of the marcumQ function. I guess, I have to ask a mathematician one day. Thanks for your answer. ---
**Author:** ![safnhd](https://avatars.discourse-cdn.com/v4/letter/s/3ab097/32.png) [@safnhd](https://community.wolfram.com/u/safnhd)\ **Post date:** [December 26, 2014, 5:20am UTC](https://community.wolfram.com/t/question-about-marcumq-function/4388/5 "2014-12-26T05:20:40Z")
Hi, Well, it seems that the reason is the mathematical definition of the function itself. Despite the fact that this property is obvious graphically, Mathematica seems unable to resolve it analytically. I hope that the program-developers will add this to its properties in a future update. Thanks for your efforts and regards. ---
**Author:** ![safnhd](https://avatars.discourse-cdn.com/v4/letter/s/3ab097/32.png) [@safnhd](https://community.wolfram.com/u/safnhd)\ **Post date:** [December 26, 2014, 8:25am UTC](https://community.wolfram.com/t/question-about-marcumq-function/4388/6 "2014-12-26T08:25:42Z")
Actually, `N[MarcumQ[1, 0, 1]]` equals 0.606531. Also, `MarcumQ[1, 1, 0]` equals 1. So obviously, the condition for a or b to be 0 is relaxed. ---
**Author:** ![sukrause](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/sukrause/32/41348_2.png) [@sukrause](https://community.wolfram.com/u/sukrause)\ **Post date:** [December 26, 2014, 9:09am UTC](https://community.wolfram.com/t/question-about-marcumq-function/4388/7 "2014-12-26T09:09:21Z")
Sorry > One of the MarcumQ properties is that > `Limit[MarcumQ[1, 3, \[Mu]],\[Mu] -> Infinity] = 1` the picture does not suggest that ![comm411586_1.PNG](https://global.discourse-cdn.com/wolfram/original/3X/e/f/eff5bf456752a6f77e576355564a39a5b7ae44c0.png) and if one is willing to accept 8 as a good approximation for \infty than the graphical result agrees with the [Fundamental theorem of calculus](http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus). * * * If you had have the \mu in the first argument of `MarcumQ` the limit would be meaningful ![comm411586_2.PNG](https://global.discourse-cdn.com/wolfram/original/3X/a/6/a69e8d8e2b0df378597bef067698e5127423a553.png) and now you have to ask yourself what \lgroup\frac{x}{a}\rgroup^{\mu-1} I\_{\mu-1}(a x) for \mu\rightarrow\infty is. For this approximations may exist. ---
**Author:** ![sukrause](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/sukrause/32/41348_2.png) [@sukrause](https://community.wolfram.com/u/sukrause)\ **Post date:** [December 26, 2014, 9:59am UTC](https://community.wolfram.com/t/question-about-marcumq-function/4388/8 "2014-12-26T09:59:11Z")
In [Connections between the Generalized MarcumQ-Function and a class of HypergeometricFunctions](http://www.dtic.upf.edu/~alozano/papers/SpecialFunctions.pdf) the equation {Q\_{m}(a,b)=1 - Q\_{1-m}(a,b)} is found. ---
**Author:** ![safnhd](https://avatars.discourse-cdn.com/v4/letter/s/3ab097/32.png) [@safnhd](https://community.wolfram.com/u/safnhd)\ **Post date:** [December 26, 2014, 9:21pm UTC](https://community.wolfram.com/t/question-about-marcumq-function/4388/9 "2014-12-26T21:21:01Z")
Hi, I disagree with most of what you said. But thanks anyways for your participation.