In mathematics , several indices are often combined into a single multi-index . From a formal point of view, a multi-index is a tuple of natural numbers .
If you generalize formulas from one variable to several variables, it usually makes sense to use multi-index notation for reasons of notation . An example would be to rewrite a power series with a variable to multiple power series . Multi-indices are often used in multidimensional analysis and theory of distributions .
Multi-index notation conventions
In this section are each tuple of natural numbers. The following conventions are usually agreed for the multi-index notation:
![{\ varvec {\ alpha}} = (\ alpha _ {1}, \ ldots, \ alpha _ {n}), \ {\ varvec {k}} = (k_ {1}, \ ldots, k_ {n} ), \ {\ boldsymbol {\ ell}} = (\ ell _ {1}, \ ldots, \ ell _ {n}) \ in \ mathbb {N} _ {0} ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e69b6ad6b1426daece837f69c8accfcb939c031)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![{\ begin {array} {ccl} {\ boldsymbol {k}} = {\ boldsymbol {\ ell}} & \ iff & k_ {1} = \ ell _ {1} \;, \; \ ldots \;, \ ; k_ {n} = \ ell _ {n} \\\\ {\ boldsymbol {k}} \ leq {\ boldsymbol {\ ell}} & \ iff & k_ {1} \ leq \ ell _ {1} \; , \; \ ldots \;, \; k_ {n} \ leq \ ell _ {n} \\\\ {\ boldsymbol {k}} + {\ boldsymbol {\ ell}} &: = & (k_ {1 } + \ ell _ {1} \;, \; \ ldots \;, \; k_ {n} + \ ell _ {n}) \\\\ {\ boldsymbol {k}}! &: = & k_ {1 }! \ cdots k_ {n}! \\\\ {{\ varvec {\ alpha}} \ choose {\ varvec {k}}} &: = & {\ frac {{\ varvec {\ alpha}}!} {({\ boldsymbol {\ alpha -k}})! \, {\ boldsymbol {k}}!}} = {\ alpha _ {1} \ choose k_ {1}} \ cdots {\ alpha _ {n} \ choose k_ {n}} \\\\ | {\ varvec {k}} | &: = & k_ {1} + \ cdots + k_ {n} \\\\ {\ varvec {x}} ^ {\ varvec {k}} &: = & x_ {1} ^ {k_ {1}} \ cdots x_ {n} ^ {k_ {n}} \\\\ {\ boldsymbol {D}} ^ {\ boldsymbol {k}} &: = & D_ {1} ^ {k_ {1}} \ cdots D_ {n} ^ {k_ {n}} \ ,, \ end {array}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/674c2edeaa8274eaf0352319c2d9813ab1336821)
where and denote a differential operator .
![{\ boldsymbol {x}} \ in \ mathbb {C} ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e69aa53663210b97c93117d6461c79b1d96dc07f)
![{\ boldsymbol {D}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff7154d73f1915f70bc06893818ef63386fb0d36)
Application examples
A multiple power series
can be briefly written as .
![\ sum _ {k_ {1} \ geq 0} \ cdots \ sum _ {k_ {n} \ geq 0} a_ {k_ {1}, \ ldots, k_ {n}} (z_ {1} -z_ {1 } ^ {o}) ^ {k_ {1}} \ cdots (z_ {n} -z_ {n} ^ {o}) ^ {k_ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1852f6f18825fed8baab14b390036b45ff0c3c0)
![\ sum _ {{\ varvec {k}} \ geq 0} a _ {\ varvec {k}} ({\ varvec {z}} - {\ varvec {z}} ^ {o}) ^ {\ varvec {k }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4944c69307b323fb4d67b7ecff3ed2ed5738bf87)
Is and are , then
and
.
![{\ displaystyle {\ boldsymbol {x}} \ in \ mathbb {R} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e24b111be77371513ddb74aba65488cb1842e05)
![{\ displaystyle {\ boldsymbol {k}}, {\ boldsymbol {m}} \ in \ mathbb {N} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a6cf07547a2abd6346bc73393c0debc7e249909)
![{\ varvec {D}} ^ {\ varvec {k}} {\ frac {{\ varvec {x}} ^ {\ varvec {m}}} {{\ varvec {m}}!}} = {\ frac {{\ varvec {x}} ^ {{\ varvec {m}} - {\ varvec {k}}}} {({\ varvec {m}} - {\ varvec {k}})!}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccf17f3ca1fef8c6273d818d2b81b5852d5e95f6)
![{\ varvec {D}} ^ {\ varvec {k}} {\ frac {| {\ varvec {x}} | ^ {m}} {m!}} = {\ frac {| {\ varvec {x} } | ^ {m- | {\ varvec {k}} |}} {(m- | {\ varvec {k}} |)!}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c49b1a19de0632b4cc782f8e0546b948adcee158)
For true
, where
is.
![- {\ varvec {1}} <{\ varvec {x}} <{\ varvec {1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9bf704fc19f91a019056e8dec0a6a54ea91156a)
![\ sum _ {| {\ varvec {k}} | \ geq 0} {\ varvec {x}} ^ {\ varvec {k}} = {\ frac {1} {({\ varvec {1}} - { \ boldsymbol {x}}) ^ {\ boldsymbol {1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c006cb6bc6685d7197ad7a178b50b5b759a7c9)
![{\ boldsymbol {1}} = (1, \ ldots, 1)](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f4840ab7abedbbb21467bcf737378542d932e63)
Are and is , then applies
or .
![{\ displaystyle {\ boldsymbol {x}}, {\ boldsymbol {y}} \ in \ mathbb {C} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08c6fdc7321eb51b40ca5710395d220c71b4b71c)
![{\ displaystyle {\ boldsymbol {m}} \ in \ mathbb {N} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da647d137e98587a27cf9e163226c54ea509e5e2)
![({\ varvec {x}} + {\ varvec {y}}) ^ {\ varvec {m}} = \ sum _ {{\ varvec {k}} \ leq {\ varvec {m}}} {{\ boldsymbol {m}} \ choose {\ boldsymbol {k}}} {\ boldsymbol {x}} ^ {\ boldsymbol {k}} {\ boldsymbol {y}} ^ {{\ boldsymbol {m}} - {\ boldsymbol {k}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1a2c41da351b244d90ae653778afa3e9078cef1)
![{\ frac {({\ varvec {x}} + {\ varvec {y}}) ^ {\ varvec {m}}} {{\ varvec {m}}!}} = \ sum _ {{\ varvec { k}} + {\ varvec {j}} = {\ varvec {m}}} {\ frac {{\ varvec {x}} ^ {\ varvec {k}}} {{\ varvec {k}}!} } {\ frac {{\ varvec {y}} ^ {\ varvec {j}}} {{\ varvec {j}}!}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/925af7cf858b822a8a9f0379d694f61077d21806)
For and is
or what can be briefly written as .
![{\ displaystyle {\ boldsymbol {x}} = (x_ {1}, \ ldots, x_ {n}) \ in \ mathbb {R} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82a287b1830ab3b9295fb399411357fcd92e0d96)
![m \ in {\ mathbb {N}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42411e85d874a733209223302bbd8d5e3ad04cb0)
![(x_ {1} + \ cdots + x_ {n}) ^ {m} = \ sum _ {k_ {1} + \ cdots + k_ {n} = m} {m \ choose k_ {1}, \ ldots, k_ {n}} x_ {1} ^ {k_ {1}} \ cdots x_ {n} ^ {k_ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79d3a743532123339885bc60afa30481f74f65ea)
![{\ frac {(x_ {1} + \ cdots + x_ {n}) ^ {m}} {m!}} = \ sum _ {k_ {1} + \ cdots + k_ {n} = m} {\ frac {x_ {1} ^ {k_ {1}}} {k_ {1}!}} \ cdots {\ frac {x_ {n} ^ {k_ {n}}} {k_ {n}!}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3212d19316dd95830066812691bb3abe9667eb0)
![{\ frac {| {\ varvec {x}} | ^ {m}} {m!}} = \ sum _ {| {\ varvec {k}} | = m} {\ frac {{\ varvec {x} } ^ {\ boldsymbol {k}}} {{\ boldsymbol {k}}!}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80f3c19f6e290bb1ad88d924e7a223807e5d8ba6)
Leibniz rule
If and are m-times continuously differentiable functions, then applies
![{\ displaystyle {\ boldsymbol {m}} \ in \ mathbb {N} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da647d137e98587a27cf9e163226c54ea509e5e2)
![f, g \ colon \ mathbb {R} ^ {n} \ to \ mathbb {R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e9e2ebf1d114db6261d32ea83c55939d8d58c3e)
![(fg) ^ {({\ varvec {m}})} = \ sum _ {{\ varvec {k}} \ leq {\ varvec {m}}} {{\ varvec {m}} \ choose {\ varvec {k}}} f ^ {({\ varvec {k}})} g ^ {({\ varvec {m}} - {\ varvec {k}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9c722cefcd83523e3765cefb32402c09068849b)
respectively
-
.
This identity is called the Leibniz rule .
And if there are m times continuously differentiable functions, then it is
![f_ {1}, \ ldots, f_ {n} \ colon \ mathbb {R} \ to \ mathbb {R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1757432a2c04fec528f10774114d61f8af8e16ef)
-
,
where is.
![{\ boldsymbol {f}} ^ {({\ boldsymbol {k}})} = (f_ {1}, \ ldots, f_ {n}) ^ {{\ big (} (k_ {1}), \ ldots , (k_ {n}) {\ big)}} = f_ {1} ^ {(k_ {1})} \ cdots f_ {n} ^ {(k_ {n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80115be37f5a56967f4bf8b52dbb80722384c4a1)
The following applies to series of multiple powers
.
![f ({\ varvec {z}}) = \ sum _ {| {\ varvec {\ ell}} | \ geq 0} a _ {\ varvec {\ ell}} \, {\ varvec {z}} ^ {\ boldsymbol {\ ell}} \;, \; g ({\ boldsymbol {z}}) = \ sum _ {| {\ boldsymbol {\ ell}} | \ geq 0} b _ {\ boldsymbol {\ ell}} \ , {\ varvec {z}} ^ {\ varvec {\ ell}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c4b3e9e5488a992a659c32fd0d9418be8099098)
![f ({\ varvec {z}}) \, g ({\ varvec {z}}) = \ sum _ {| {\ varvec {\ ell}} | \ geq 0} \ left (\ sum _ {{\ boldsymbol {k}} + {\ boldsymbol {j}} = {\ boldsymbol {\ ell}}} a _ {\ boldsymbol {k}} \, b _ {\ boldsymbol {j}} \ right) {\ boldsymbol {z} } ^ {\ boldsymbol {\ ell}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db72b9f06c969498a2b14f1c341859ad313ef9cb)
If power series are a variable, then where is.
![f_ {1} (z) = \ sum _ {\ ell = 0} ^ {\ infty} a_ {1 \ ell} z ^ {\ ell} \;, \; \ ldots \;, \; f_ {n} (z) = \ sum _ {\ ell = 0} ^ {\ infty} a_ {n \ ell} z ^ {\ ell}](https://wikimedia.org/api/rest_v1/media/math/render/svg/756b981129ac343f461c94a1b357c83ceeaf804c)
![f_ {1} (z) \ cdots f_ {n} (z) = \ sum _ {\ ell = 0} ^ {\ infty} \ left (\ sum _ {| {\ boldsymbol {k}} | = \ ell } a _ {\ boldsymbol {k}} \ right) z ^ {\ ell}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98fd2589f9e1623a6564f365f9eb126f388325d8)
![a _ {\ boldsymbol {k}} = a_ {1k_ {1}} \ cdots a_ {nk_ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47604438386aa3670369857bdb2b3a308315af10)
For true .
![{\ displaystyle {\ boldsymbol {z}} = (z_ {1}, ..., z_ {n}) \ in \ mathbb {C} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5468fea65975b9b34a03c61b654b1b1a4c97488)
![e ^ {z_ {1} + ... + z_ {n}} = \ sum _ {{\ varvec {k}} \ geq 0} {\ frac {{\ varvec {z}} ^ {\ varvec {k }}} {{\ boldsymbol {k}}!}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b39df10e431c99a52e72629161c527ac3e171deb)
If and are all components of
absolute value , then applies .
![{\ displaystyle {\ varvec {\ alpha}}, {\ varvec {x}} \ in \ mathbb {C} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15cecd27195d577b1c85e4403368181c5ff3e920)
![{\ boldsymbol {x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/606b7680d510560a505937143775ea80fa958051)
![<1 \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/24386adb5cf7b50987ceaaf73c0843a1423194b4)
![({\ varvec {1}} + {\ varvec {x}}) ^ {\ varvec {\ alpha}} = \ sum _ {| {\ varvec {k}} | \ geq 0} {{\ varvec {\ alpha}} \ choose {\ varvec {k}}} \, {\ varvec {x}} ^ {\ varvec {k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9aa148cb6d0e71453c27006bcf543865285c873)
Is and are , so applies
.
![{\ displaystyle {\ boldsymbol {m}} \ in \ mathbb {N} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da647d137e98587a27cf9e163226c54ea509e5e2)
![{\ displaystyle {\ varvec {\ alpha}}, {\ varvec {\ beta}} \ in \ mathbb {C} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88ace3e869b49b1a466a5055b0b2cb3cb19e4dc9)
![{{\ varvec {\ alpha}} + {\ varvec {\ beta}} \ choose {\ varvec {m}}} = \ sum _ {{\ varvec {k}} \ leq {\ varvec {m}}} {{\ varvec {\ alpha}} \ choose {\ varvec {k}}} {{\ varvec {\ beta}} \ choose {\ varvec {m}} - {\ varvec {k}}} = \ sum _ {{\ varvec {k}} + {\ varvec {j}} = {\ varvec {m}}} {{\ varvec {\ alpha}} \ choose {\ varvec {k}}} {{\ varvec {\ beta}} \ choose {\ boldsymbol {j}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/867a62067e945a8a3f830910c8e4862c2a48bcd4)
Is and so is true
.
![m \ in {\ mathbb {N}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42411e85d874a733209223302bbd8d5e3ad04cb0)
![{\ displaystyle {\ boldsymbol {\ alpha}} = (\ alpha _ {1}, ..., \ alpha _ {n}) \ in \ mathbb {C} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a991e7c1c25d320baa64f54434cfabe4161e3099)
![{| {\ varvec {\ alpha}} | \ choose m} = \ sum _ {| {\ varvec {k}} | = m} {{\ varvec {\ alpha}} \ choose {\ varvec {k}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d867468ee43760fe26e312b13f931363a553766)
Cauchy's integral formula
Cauchy's integral formula can be divided into several variables
![{\ frac {D ^ {\ boldsymbol {k}} f (z_ {1}, \ ldots, z_ {n})} {{\ boldsymbol {k}}!}} = {\ frac {1} {(2nd \ pi i) ^ {n}}} \ oint _ {\ partial U_ {n}} \ cdots \ oint _ {\ partial U_ {1}} {\ frac {f (\ xi _ {1}, \ ldots, \ xi _ {n})} {(\ xi _ {1} -z_ {1}) ^ {k_ {1} +1} \ cdots (\ xi _ {n} -z_ {n}) ^ {k_ { n} +1}}} d \ xi _ {1} \ cdots d \ xi _ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95823dcae144747c4eb99a73596bbcc3fc8c2dbe)
write briefly as
-
,
where should be. The estimate where is also applies .
![\ partial {\ boldsymbol {U}} = \ partial U_ {1} \ times \ cdots \ times \ partial U_ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/511614602d807c4e8d9de0ddba0a613d7b48af18)
![| a _ {\ varvec {k}} | \ leq {\ tfrac {M} {{\ varvec {r}} ^ {\ varvec {k}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34058a3f32c580afc5489f2be6a0c35343d111f6)
![\ textstyle M = \ max _ {{\ varvec {\ xi}} \ in \ partial {\ varvec {U}}} | f ({\ varvec {k}}) |](https://wikimedia.org/api/rest_v1/media/math/render/svg/43d40b7a5c76ca8bcffb673303af65d06307a729)
Taylor series
If it is an analytic function or a holomorphic mapping , this function can be converted into a Taylor series![{\ displaystyle f \ colon \ mathbb {R} ^ {n} \ to \ mathbb {R}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6991fd928a18a993d719e93137af6240a9b6545a)
![f ({\ varvec {z}}) = \ sum _ {| {\ varvec {k}} | \ geq 0} {\ frac {D ^ {\ varvec {k}} f ({\ varvec {z}} ^ {o})} {{\ varvec {k}}!}} (zz ^ {o}) ^ {\ varvec {k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f810dd9f99a0e368c78ce4d9fbf6cb41102d3695)
develop, where is a multi-index.
![{\ boldsymbol {k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5007edd8c173fdfa24afccd09ea3ba5fb2a12a9)
Hurwitz identity
For with and applies
.
![x, y \ in {\ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75c3842d5b9fc0aac819c2e05942151efd45708f)
![x \ neq 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/35a455db7b2aab1b0e72ccbc7385e4424e2372e5)
![{\ displaystyle {\ boldsymbol {a}} = (a_ {1}, ..., a_ {n}) \ in \ mathbb {C} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4507137087263d1373060329397b1303c3845b5)
![(x + y) ^ {n} = \ sum _ {{\ varvec {0}} \ leq {\ varvec {k}} \ leq {\ varvec {1}}} x \, (x + {\ varvec {a }} \ cdot {\ varvec {k}}) ^ {| {\ varvec {k}} | -1} \, (y - {\ varvec {a}} \ cdot {\ varvec {k}}) ^ { n- | {\ boldsymbol {k}} |}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e46e3119462e2eaf3896b9791330be8b59fa734b)
This generalizes the Abelian identity .
![(x + y) ^ {n} = \ sum _ {k = 0} ^ {n} {n \ choose k} \, x \, (x + ak) ^ {k-1} \, (y-ak ) ^ {nk}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ebcf44f2f265e3bfacf8ebb42a7d615f3a0fdee)
The latter is obtained in the case .
![{\ boldsymbol {a}} = (a, a, ..., a)](https://wikimedia.org/api/rest_v1/media/math/render/svg/efefbd64fe9a671b681b09c520e509a12e125595)
literature
- Otto Forster: Analysis. Volume 2: differential calculus in R n . Ordinary differential equations. 7th improved edition. Vieweg + Teubner, Wiesbaden 2006, ISBN 3-8348-0250-6 ( Vieweg study. Basic course in mathematics ).
-
Konrad Königsberger : Analysis. Volume 2. 3rd revised edition. Springer-Verlag, Berlin et al. 2000, ISBN 3-540-66902-7 .