In mathematics , various special functions are called a dilogarithm . The classic dilogarithm is a special case of the polylogarithm .
Classic dilogarithm
Values of the classical dilogarithm on the real axis. (The imaginary part is identically zero there.)
The classic Dilogarithm for complex numbers with defined by the power series![z](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98)
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.
It can be continued through analytical continuation on :
![{\ displaystyle \ mathbb {C} \ setminus \ left [1, \ infty \ right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab291bc8bd82894a5a6410d8a956e0909cd6d128)
![\ operatorname {Li} _ {2} (z) = - \ int _ {0} ^ {z} {{\ frac {\ log (1-t)} {t}}} \, {\ mathrm {d} } t.](https://wikimedia.org/api/rest_v1/media/math/render/svg/68e29bebf8724316f6e4d4c0541abeaaab544abb)
(This must be integrated along a path in .)
![{\ displaystyle \ mathbb {C} \ setminus \ left [1, \ infty \ right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab291bc8bd82894a5a6410d8a956e0909cd6d128)
Bloch-Wigner dilogarithm
The Bloch-Wigner dilogarithm is defined for by
![{\ displaystyle z \ in \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/169fae60c23a2027ece2aa7fd4b5047492887e91)
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.
It's well-defined and steady, even in .
![\ left [1, \ infty \ right]](https://wikimedia.org/api/rest_v1/media/math/render/svg/af21cee37c311938eaee99380143575c8a7c3678)
It is analytical in , in 0 and 1 it has singularities of type .
![{\ displaystyle \ mathbb {C} \ setminus \ left \ {0.1 \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2793793545a5d0dd193f4decc0489339d4c43eb)
![r \ log (r)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6ec7613964b7f7126d4011b2569be891255350a)
Rogers dilogarithm
The Rogers dilogarithm is defined by
![L (x) = {\ frac {6} {\ pi ^ {2}}} \ left (\ operatorname {Li} _ {2} (x) + {\ frac {1} {2}} \ log (x ) \ log (1-x) \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/d77f00b269c47f19701c1f11e75e1915fa4c7ce7)
for .
![0 <x <1](https://wikimedia.org/api/rest_v1/media/math/render/svg/a440e33e5630b5f22cd3ca24cfdf85f56965ac8f)
Another common definition is
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.
This depends on the former via
![R (x) = {\ frac {\ pi ^ {2}} {6}} (L (x) -1)](https://wikimedia.org/api/rest_v1/media/math/render/svg/887b8b16315c458138a1794f6b2156c9dcc200df)
together.
One can (discontinuously) continue on completely through and
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![\ mathbb {R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc)
![R (1) = 0, R (0) = - {\ frac {\ pi ^ {2}} {6}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/801b58ecc4ebaae318f23bedf6007229d3a39365)
![R (x) = \ left \ {{\ begin {array} {c} -R (1 / x) \ {\ mbox {for}} \ x> 1 \\ - R (x / (x-1)) \ {\ mbox {for}} \ x <0 \ end {array}} \ right \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80477167d675801b9c69b7c7ffd9f7cbf7fcc893)
Elliptical dilogarithm
Let be an elliptic curve defined over . It can be parameterized by
means of a grid using the Weierstrasse schen function![E.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b)
![\ mathbb {Q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a)
![\ Lambda = \ left \ {1, \ tau \ right \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6ade31f20ab7ed93aff33e9ef394a98fc8ad30d)
![{\ displaystyle \ mathbb {C} / \ Lambda \ rightarrow E (\ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf7b1b5a698aea6e3dc5e0f9a331e97c2406648e)
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mod .![\ Lambda \ mapsto (p (u), p ^ {\ prime} (u))](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f837f9f5fe7ef65446c9083e21f88a85b8ffa33)
The elliptical dilogarithm is then defined by
![{\ displaystyle D ^ {E} \ colon E (\ mathbb {C}) \ rightarrow \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f17315c986967db69488c6127ed134ddd56e41f)
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,
where denotes the Bloch-Wigner dilogarithm.
![D_ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41b3839c40bd06e3dfea10798dfab41a905af256)
The elliptical dilogarithm agrees with the value of the L-function except for rational multiples of .
![\pi](https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a)
![{\ displaystyle L (E, 2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cc66cae1ac5f74f8ac86c3633ea43a0223cfe36)
Special values
Classic dilogarithm
For the following numbers, and can be represented in closed form:
![z](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98)
![\ operatorname {Li} _ {2} (z)](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0fc479bb3abce79e8a0ed0566ba71e182687907)
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,
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,
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,
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.
With the sixth root of unity and the Gieseking constant one also has
![V_0 = 1 {,} 0149 \ ldots](https://wikimedia.org/api/rest_v1/media/math/render/svg/4deb9aee8f58d069dd478c82c2f2b04e5d038f30)
![\ operatorname {Li} _2 (\ omega) = \ frac {\ pi ^ 2} {36} + V_0i, \ qquad \ operatorname {Li} _2 (\ omega ^ 2) = - \ frac {\ pi ^ 2} { 18} + \ frac {2} {3} V_0i](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc4742e92f67a90e24d36fced210e0bbecba2b6f)
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.
Bloch-Wigner dilogarithm
So far, values of the Bloch-Wigner dilogarithm can only be calculated numerically and only a few algebraic relations are known between values of the Bloch-Wigner dilogarithm. An assumption by John Milnor states that :
![N \ geq 3](https://wikimedia.org/api/rest_v1/media/math/render/svg/80c4b031802faaf0e670460e78725391931e7329)
- the numbers for and are linearly independent over .
![D_ {2} (e ^ {{2 \ pi i {\ frac {j} {N}}}})](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e906f80448e2dcf56f7585a2bbe1638b702896b)
![0 <j <{\ frac {N} {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8ea5ac010659656eb254f69f1619770ab4a2e3)
![\ operatorname {ggT} (j, N) = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a7b49d590cddd0cab08536f1db879a0efb8aae4)
![\ mathbb {Q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a)
Rogers dilogarithm
There are numerous algebraic identities between values of in rational or algebraic arguments. Examples of special values are
![L.](https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8)
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.
With the sixth root of unity and the Gieseking constant one has
![V_ {0} = 1.0149 ...](https://wikimedia.org/api/rest_v1/media/math/render/svg/017b01dcfc8430787740083d85e2924cde30611a)
![R (\ omega) = - \ frac {\ pi ^ 2} {12} + V_0i, \ qquad R (\ omega ^ 2) = - \ frac {\ pi ^ 2} {6} + (\ frac {2} {3} V_0 + \ frac {1} {6} \ ln (3) \ pi) i](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9cb73c3fcba9cd398adc2610f9eb4d5493680e1)
![R (1+ \ omega) = (\ frac {2} {3} V_0 + \ frac {1} {6} \ ln (3) \ pi) i, \ qquad R (\ frac {1} {1+ \ omega }) = - \ frac {\ pi ^ 2} {12} - \ frac {1} {8} \ ln (3) + \ frac {1} {8} (\ ln (3)) ^ 2 + (- \ frac {2} {3} V_0 + \ frac {1} {12} \ ln (3) \ pi) i](https://wikimedia.org/api/rest_v1/media/math/render/svg/d11cb16e6be8c365ebe145bb9f01bc5ddf7f6359)
Functional equations
Classic dilogarithm
The classical dilogarithm suffices for numerous functional equations, for example
![\ operatorname {Li} _ {2} (z) + \ operatorname {Li} _ {2} (- z) = {\ frac {1} {2}} \ operatorname {Li} _ {2} (z ^ { 2})](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf8318d731f0e865f9bfaa449f320cbbdf66030a)
![\ operatorname {Li} _ {2} (1-z) + \ operatorname {Li} _ {2} \ left (1 - {\ frac {1} {z}} \ right) = - {\ frac {\ ln ^ {2} z} {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/370f80c1dd2d12806be30dcabaeab6999cb37745)
![\ operatorname {Li} _ {2} (z) + \ operatorname {Li} _ {2} (1-z) = {\ frac {{\ pi} ^ {2}} {6}} - \ ln z \ cdot \ ln (1-z)](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bbcd6238218c0c907b5f0fed27b1dd1f2338e1e)
![\ operatorname {Li} _ {2} (- z) - \ operatorname {Li} _ {2} (1-z) + {\ frac {1} {2}} \ operatorname {Li} _ {2} (1 -z ^ {2}) = - {\ frac {{\ pi} ^ {2}} {12}} - \ ln z \ cdot \ ln (z + 1)](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b8d1720c0458890ff1dbd1ce930469000d4d49a)
![\ operatorname {Li} _ {2} (z) + \ operatorname {Li} _ {2} \ left ({\ frac {1} {z}} \ right) = - {\ frac {\ pi ^ {2} } {6}} - {\ frac {1} {2}} \ ln ^ {2} (- z)](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd872721734d1f4eb2a1eb480a7e25728c418057)
Bloch-Wigner dilogarithm
The Bloch-Wigner dilogarithm is sufficient for identities
![\ operatorname {D} _ {2} (z) = \ operatorname {D} _ {2} \ left (1 - {\ frac {1} {z}} \ right) = \ operatorname {D} _ {2} \ left ({\ frac {1} {1-z}} \ right) = - \ operatorname {D} _ {2} \ left ({\ frac {1} {z}} \ right) = - \ operatorname { D} _ {2} (1-z) = - \ operatorname {D} _ {2} \ left ({\ frac {-z} {1-z}} \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/63f54b25b02ba529c32d797865bea5b5378d811e)
and the 5-term relation
![\ operatorname {D} _ {2} (x) + \ operatorname {D} _ {2} (y) + \ operatorname {D} _ {2} \ left ({\ frac {1-x} {1-xy }} \ right) + \ operatorname {D} _ {2} (1-xy) + \ operatorname {D} _ {2} \ left ({\ frac {1-y} {1-xy}} \ right) = 0.](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdb5085a5abe7ec3b9aa6f1150293a2a90c36338)
Rogers dilogarithm
The Rogers dilogarithm fulfills the relationship
![L (x) + L (1-x) = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/237d767cdfcd0b574c8a24b7a54ba550d6b8a505)
and Abel's functional equation
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.
For one has
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
![R (x) + R (1-x) = - {\ frac {\ pi ^ {2}} {6}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45f948a9498a98d72c657e7a45b65204a1a958b0)
and the 5-term relation
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,
in particular is a well-defined function on the Bloch group .
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
See also
Web links
Individual evidence
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^ K 2 and L-functions of elliptic curves