Natural logarithm

Symbol: Log —

\log(z)

— Natural logarithm

The principal branch of the natural logarithm

\log(z)

is a function of one complex variable

z

.

It has a branch point singularity at

z = 0

and a branch cut on

\left(-\infty, 0\right]

where the value on

\left(-\infty, 0\right)

is taken to be continuous with the upper half plane.

The following table lists all conditions such that Log(z) is defined in Fungrim.

Domain	Codomain
Numbers
$z \in \left\{1\right\}$	$\log(z) \in \left\{0\right\}$
$z \in \left(0, \infty\right)$	$\log(z) \in \mathbb{R}$
$z \in \mathbb{C} \setminus \left\{0\right\}$	$\log(z) \in \mathbb{C}$
Infinities
$z \in \left\{\infty\right\}$	$\log(z) \in \left\{\infty\right\}$
Formal power series
$z \in \mathbb{Q}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] z = 1$	$\log(z) \in \mathbb{Q}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] \log(z) = 0$
$z \in \mathbb{R}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] z \in \left(0, \infty\right)$	$\log(z) \in \mathbb{R}[[x]]$
$z \in \mathbb{C}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] z \ne 0$	$\log(z) \in \mathbb{C}[[x]]$

Table data:

\left(P, Q\right)

such that

\left(P\right) \;\implies\; \left(Q\right)

Definitions:

Fungrim symbol	Notation	Short description
`Log`	$\log(z)$	Natural logarithm
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`OpenInterval`	$\left(a, b\right)$	Open interval
`RR`	$\mathbb{R}$	Real numbers
`CC`	$\mathbb{C}$	Complex numbers
`PowerSeries`	$K[[x]]$	Formal power series
`QQ`	$\mathbb{Q}$	Rational numbers

Source code for this entry:

Entry(ID("ed210c"),
    SymbolDefinition(Log, Log(z), "Natural logarithm"),
    Description("The principal branch of the natural logarithm", Log(z), "is a function of one complex variable", z, "."),
    Description("It has a branch point singularity at", Equal(z, 0), "and a branch cut on", OpenClosedInterval(Neg(Infinity), 0), "where the value on", OpenInterval(Neg(Infinity), 0), "is taken to be continuous with the upper half plane."),
    Description("The following table lists all conditions such that", SourceForm(Log(z)), "is defined in Fungrim."),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(TableSection("Numbers"), Tuple(Element(z, Set(1)), Element(Log(z), Set(0))), Tuple(Element(z, OpenInterval(0, Infinity)), Element(Log(z), RR)), Tuple(Element(z, SetMinus(CC, Set(0))), Element(Log(z), CC)), TableSection("Infinities"), Tuple(Element(z, Set(Infinity)), Element(Log(z), Set(Infinity))), TableSection("Formal power series"), Tuple(And(Element(z, PowerSeries(QQ, x)), Equal(SeriesCoefficient(z, x, 0), 1)), And(Element(Log(z), PowerSeries(QQ, x)), Equal(SeriesCoefficient(Log(z), x, 0), 0))), Tuple(And(Element(z, PowerSeries(RR, x)), Element(SeriesCoefficient(z, x, 0), OpenInterval(0, Infinity))), And(Element(Log(z), PowerSeries(RR, x)))), Tuple(And(Element(z, PowerSeries(CC, x)), NotEqual(SeriesCoefficient(z, x, 0), 0)), And(Element(Log(z), PowerSeries(CC, x)))))))

4fe0ff

Image: X-ray of

\log(z)

on

z \in \left[-3, 3\right] + \left[-3, 3\right] i

Download: png (small) — png (medium) — png (large) — pdf (small) — pdf (medium/large) — svg (small) — svg (medium/large)

An X-ray plot illustrates the geometry of a complex analytic function

f(z)

. Thick black curves show where

\operatorname{Im}\!\left(f(z)\right) = 0

(the function is pure real). Thick red curves show where

\operatorname{Re}\!\left(f(z)\right) = 0

(the function is pure imaginary). Points where black and red curves intersect are zeros or poles. Magnitude level curves

\left|f(z)\right| = C

are rendered as thin gray curves, with brighter shades corresponding to larger

C

. Blue lines show branch cuts. The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line. Yellow is used to highlight important regions.

Definitions:

Fungrim symbol	Notation	Short description
`Log`	$\log(z)$	Natural logarithm
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`ConstI`	$i$	Imaginary unit
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("4fe0ff"),
    Image(Description("X-ray of", Log(z), "on", Element(z, Add(ClosedInterval(-3, 3), Mul(ClosedInterval(-3, 3), ConstI)))), ImageSource("xray_log")),
    Description("An X-ray plot illustrates the geometry of a complex analytic function", f(z), ".", "Thick black curves show where", Equal(Im(f(z)), 0), "(the function is pure real).", "Thick red curves show where", Equal(Re(f(z)), 0), "(the function is pure imaginary).", "Points where black and red curves intersect are zeros or poles.", "Magnitude level curves", Equal(Abs(f(z)), C), "are rendered as thin gray curves, with brighter shades corresponding to larger", C, ".", "Blue lines show branch cuts.", "The value of the function is continuous with the branch cut on the side indicated with a solid line, and discontinuous on the side indicated with a dashed line.", "Yellow is used to highlight important regions."))

07731b

\log(1) = 0

TeX:

\log(1) = 0

Definitions:

Fungrim symbol	Notation	Short description
`Log`	$\log(z)$	Natural logarithm

Source code for this entry:

Entry(ID("07731b"),
    Formula(Equal(Log(1), 0)))

699c83

\log(e) = 1

TeX:

\log(e) = 1

Definitions:

Fungrim symbol	Notation	Short description
`Log`	$\log(z)$	Natural logarithm
`ConstE`	$e$	The constant e (2.718...)

Source code for this entry:

Entry(ID("699c83"),
    Formula(Equal(Log(ConstE), 1)))

d496b8

Table of

\log(n)

to 50 digits for

1 \le n \le 50

$n$	$\log(n) \; (\text{nearest } 50 \text{D})$
1	0
2	0.69314718055994530941723212145817656807550013436026
3	1.0986122886681096913952452369225257046474905578227
4	1.3862943611198906188344642429163531361510002687205
5	1.6094379124341003746007593332261876395256013542685
6	1.7917594692280550008124773583807022727229906921830
7	1.9459101490553133051053527434431797296370847295819
8	2.0794415416798359282516963643745297042265004030808
9	2.1972245773362193827904904738450514092949811156455
10	2.3025850929940456840179914546843642076011014886288
11	2.3978952727983705440619435779651292998217068539374
12	2.4849066497880003102297094798388788407984908265433
13	2.5649493574615367360534874415653186048052679447602
14	2.6390573296152586145225848649013562977125848639421
15	2.7080502011022100659960045701487133441730919120913
16	2.7725887222397812376689284858327062723020005374410
17	2.8332133440562160802495346178731265355882030125857
18	2.8903717578961646922077225953032279773704812500058
19	2.9444389791664404600090274318878535372373792612991
20	2.9957322735539909934352235761425407756766016229890
21	3.0445224377234229965005979803657054342845752874046
22	3.0910424533583158534791756994233058678972069882977
23	3.1354942159291496908067528318101961184423803148404
24	3.1780538303479456196469416012970554088739909609035
25	3.2188758248682007492015186664523752790512027085370
26	3.2580965380214820454707195630234951728807680791205
27	3.2958368660043290741857357107675771139424716734682
28	3.3322045101752039239398169863595328657880849983024
29	3.3672958299864740271832720323619116054945129139227
30	3.4011973816621553754132366916068899122485920464515
31	3.4339872044851462459291643245423572104499389304806
32	3.4657359027997265470861606072908828403775006718013
33	3.4965075614664802354571888148876550044691974117602
34	3.5263605246161613896667667393313031036637031469460
35	3.5553480614894136797061120766693673691626860838504
36	3.5835189384561100016249547167614045454459813843660
37	3.6109179126442244443680956710314471639000775871676
38	3.6375861597263857694262595533460301053128793956594
39	3.6635616461296464274487326784878443094527585025830
40	3.6888794541139363028524556976007173437521017573493
41	3.7135720667043078038667633730374075883764104693993
42	3.7376696182833683059178301018238820023600754217649
43	3.7612001156935624234728425133458470355591361848816
44	3.7841896339182611628964078208814824359727071226579
45	3.8066624897703197573912498070712390488205824699140
46	3.8286413964890950002239849532683726865178804492007
47	3.8501476017100585868209506697721737088960505020202
48	3.8712010109078909290641737227552319769494910952638
49	3.8918202981106266102107054868863594592741694591637
50	3.9120230054281460586187507879105518471267028428973

Definitions:

Fungrim symbol	Notation	Short description
`Log`	$\log(z)$	Natural logarithm

Source code for this entry:

Entry(ID("d496b8"),
    Description("Table of", Log(n), "to 50 digits for", LessEqual(1, n, 50)),
    Table(Var(n), TableValueHeadings(n, NearestDecimal(Log(n), 50)), TableSplit(1), List(Tuple(1, Decimal("0")), Tuple(2, Decimal("0.69314718055994530941723212145817656807550013436026")), Tuple(3, Decimal("1.0986122886681096913952452369225257046474905578227")), Tuple(4, Decimal("1.3862943611198906188344642429163531361510002687205")), Tuple(5, Decimal("1.6094379124341003746007593332261876395256013542685")), Tuple(6, Decimal("1.7917594692280550008124773583807022727229906921830")), Tuple(7, Decimal("1.9459101490553133051053527434431797296370847295819")), Tuple(8, Decimal("2.0794415416798359282516963643745297042265004030808")), Tuple(9, Decimal("2.1972245773362193827904904738450514092949811156455")), Tuple(10, Decimal("2.3025850929940456840179914546843642076011014886288")), Tuple(11, Decimal("2.3978952727983705440619435779651292998217068539374")), Tuple(12, Decimal("2.4849066497880003102297094798388788407984908265433")), Tuple(13, Decimal("2.5649493574615367360534874415653186048052679447602")), Tuple(14, Decimal("2.6390573296152586145225848649013562977125848639421")), Tuple(15, Decimal("2.7080502011022100659960045701487133441730919120913")), Tuple(16, Decimal("2.7725887222397812376689284858327062723020005374410")), Tuple(17, Decimal("2.8332133440562160802495346178731265355882030125857")), Tuple(18, Decimal("2.8903717578961646922077225953032279773704812500058")), Tuple(19, Decimal("2.9444389791664404600090274318878535372373792612991")), Tuple(20, Decimal("2.9957322735539909934352235761425407756766016229890")), Tuple(21, Decimal("3.0445224377234229965005979803657054342845752874046")), Tuple(22, Decimal("3.0910424533583158534791756994233058678972069882977")), Tuple(23, Decimal("3.1354942159291496908067528318101961184423803148404")), Tuple(24, Decimal("3.1780538303479456196469416012970554088739909609035")), Tuple(25, Decimal("3.2188758248682007492015186664523752790512027085370")), Tuple(26, Decimal("3.2580965380214820454707195630234951728807680791205")), Tuple(27, Decimal("3.2958368660043290741857357107675771139424716734682")), Tuple(28, Decimal("3.3322045101752039239398169863595328657880849983024")), Tuple(29, Decimal("3.3672958299864740271832720323619116054945129139227")), Tuple(30, Decimal("3.4011973816621553754132366916068899122485920464515")), Tuple(31, Decimal("3.4339872044851462459291643245423572104499389304806")), Tuple(32, Decimal("3.4657359027997265470861606072908828403775006718013")), Tuple(33, Decimal("3.4965075614664802354571888148876550044691974117602")), Tuple(34, Decimal("3.5263605246161613896667667393313031036637031469460")), Tuple(35, Decimal("3.5553480614894136797061120766693673691626860838504")), Tuple(36, Decimal("3.5835189384561100016249547167614045454459813843660")), Tuple(37, Decimal("3.6109179126442244443680956710314471639000775871676")), Tuple(38, Decimal("3.6375861597263857694262595533460301053128793956594")), Tuple(39, Decimal("3.6635616461296464274487326784878443094527585025830")), Tuple(40, Decimal("3.6888794541139363028524556976007173437521017573493")), Tuple(41, Decimal("3.7135720667043078038667633730374075883764104693993")), Tuple(42, Decimal("3.7376696182833683059178301018238820023600754217649")), Tuple(43, Decimal("3.7612001156935624234728425133458470355591361848816")), Tuple(44, Decimal("3.7841896339182611628964078208814824359727071226579")), Tuple(45, Decimal("3.8066624897703197573912498070712390488205824699140")), Tuple(46, Decimal("3.8286413964890950002239849532683726865178804492007")), Tuple(47, Decimal("3.8501476017100585868209506697721737088960505020202")), Tuple(48, Decimal("3.8712010109078909290641737227552319769494910952638")), Tuple(49, Decimal("3.8918202981106266102107054868863594592741694591637")), Tuple(50, Decimal("3.9120230054281460586187507879105518471267028428973")))))

c331da

\log(i) = \frac{\pi i}{2}

TeX:

\log(i) = \frac{\pi i}{2}

Definitions:

Fungrim symbol	Notation	Short description
`Log`	$\log(z)$	Natural logarithm
`ConstI`	$i$	Imaginary unit
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("c331da"),
    Formula(Equal(Log(ConstI), Div(Mul(Pi, ConstI), 2))))

2f1f7b

\log(-1) = \pi i

TeX:

\log(-1) = \pi i

Definitions:

Fungrim symbol	Notation	Short description
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("2f1f7b"),
    Formula(Equal(Log(-1), Mul(Pi, ConstI))))

d87f6e

{e}^{\log(z)} = z

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

{e}^{\log(z)} = z

z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`Log`	$\log(z)$	Natural logarithm
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("d87f6e"),
    Formula(Equal(Exp(Log(z)), z)),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

4c1e1e

\log\!\left({e}^{z}\right) = z

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Im}(z) \in \left(-\pi, \pi\right]

TeX:

\log\!\left({e}^{z}\right) = z

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Im}(z) \in \left(-\pi, \pi\right]

Definitions:

Fungrim symbol	Notation	Short description
`Log`	$\log(z)$	Natural logarithm
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers
`Im`	$\operatorname{Im}(z)$	Imaginary part
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Pi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("4c1e1e"),
    Formula(Equal(Log(Exp(z)), z)),
    Variables(z),
    Assumptions(And(Element(z, CC), Element(Im(z), OpenClosedInterval(Neg(Pi), Pi)))))

a3a253

\log\!\left({e}^{z}\right) = z - 2 \pi i \left\lceil \frac{\operatorname{Im}(z)}{2 \pi} - \frac{1}{2} \right\rceil

Assumptions:

z \in \mathbb{C}

TeX:

\log\!\left({e}^{z}\right) = z - 2 \pi i \left\lceil \frac{\operatorname{Im}(z)}{2 \pi} - \frac{1}{2} \right\rceil

z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Log`	$\log(z)$	Natural logarithm
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Im`	$\operatorname{Im}(z)$	Imaginary part
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("a3a253"),
    Formula(Equal(Log(Exp(z)), Sub(z, Mul(Mul(Mul(2, Pi), ConstI), Ceil(Sub(Div(Im(z), Mul(2, Pi)), Div(1, 2))))))),
    Variables(z),
    Assumptions(Element(z, CC)))

c43533

\log(z) = \log\!\left(\left|z\right|\right) + \arg(z) i

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

\log(z) = \log\!\left(\left|z\right|\right) + \arg(z) i

z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Log`	$\log(z)$	Natural logarithm
`Abs`	$\left\|z\right\|$	Absolute value
`Arg`	$\arg(z)$	Complex argument
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("c43533"),
    Formula(Equal(Log(z), Add(Log(Abs(z)), Mul(Arg(z), ConstI)))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

f67fa2

\log\!\left(c z\right) = \log(c) + \log(z)

Assumptions:

c \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

\log\!\left(c z\right) = \log(c) + \log(z)

c \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Log`	$\log(z)$	Natural logarithm
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("f67fa2"),
    Formula(Equal(Log(Mul(c, z)), Add(Log(c), Log(z)))),
    Variables(c, z),
    Assumptions(And(Element(c, OpenInterval(0, Infinity)), Element(z, SetMinus(CC, Set(0))))))

4538ba

\log(z) \text{ is holomorphic on } z \in \mathbb{C} \setminus \left(-\infty, 0\right]

TeX:

\log(z) \text{ is holomorphic on } z \in \mathbb{C} \setminus \left(-\infty, 0\right]

Definitions:

Fungrim symbol	Notation	Short description
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	Holomorphic predicate
`Log`	$\log(z)$	Natural logarithm
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("4538ba"),
    Formula(IsHolomorphic(Log(z), ForElement(z, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0))))))

c464e3

\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C} \cup \left\{{\tilde \infty}\right\}} \log(z) = \left\{\right\}

TeX:

\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C} \cup \left\{{\tilde \infty}\right\}} \log(z) = \left\{\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Log`	$\log(z)$	Natural logarithm
`CC`	$\mathbb{C}$	Complex numbers
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity

Source code for this entry:

Entry(ID("c464e3"),
    Formula(Equal(Poles(Log(z), ForElement(z, Union(CC, Set(UnsignedInfinity)))), Set())))

ddc8a1

\operatorname{EssentialSingularities}\!\left(\log(z), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}

TeX:

\operatorname{EssentialSingularities}\!\left(\log(z), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Log`	$\log(z)$	Natural logarithm
`CC`	$\mathbb{C}$	Complex numbers
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity

Source code for this entry:

Entry(ID("ddc8a1"),
    Formula(Equal(EssentialSingularities(Log(z), z, Union(CC, Set(UnsignedInfinity))), Set())))

940c48

\operatorname{BranchPoints}\!\left(\log(z), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{{\tilde \infty}, 0\right\}

TeX:

\operatorname{BranchPoints}\!\left(\log(z), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{{\tilde \infty}, 0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Log`	$\log(z)$	Natural logarithm
`CC`	$\mathbb{C}$	Complex numbers
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity

Source code for this entry:

Entry(ID("940c48"),
    Formula(Equal(BranchPoints(Log(z), z, Union(CC, Set(UnsignedInfinity))), Set(UnsignedInfinity, 0))))

b5ded1

\operatorname{BranchCuts}\!\left(\log(z), z, \mathbb{C}\right) = \left\{\left(-\infty, 0\right]\right\}

TeX:

\operatorname{BranchCuts}\!\left(\log(z), z, \mathbb{C}\right) = \left\{\left(-\infty, 0\right]\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Log`	$\log(z)$	Natural logarithm
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("b5ded1"),
    Formula(Equal(BranchCuts(Log(z), z, CC), Set(OpenClosedInterval(Neg(Infinity), 0)))))

ed6590

\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, \log(z) = \log\!\left(-b\right) + \pi i

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Im}(a) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Im}(b) < 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) \operatorname{Im}(b) - \operatorname{Re}(b) \operatorname{Im}(a) > 0

TeX:

\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, \log(z) = \log\!\left(-b\right) + \pi i

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Im}(a) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Im}(b) < 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) \operatorname{Im}(b) - \operatorname{Re}(b) \operatorname{Im}(a) > 0

Definitions:

Fungrim symbol	Notation	Short description
`AnalyticContinuation`	$\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, f(z)$	Analytic continuation
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("ed6590"),
    Formula(Equal(AnalyticContinuation(Log(z), For(z, a, b)), Add(Log(Neg(b)), Mul(Pi, ConstI)))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC), Greater(Im(a), 0), Less(Im(b), 0), Greater(Sub(Mul(Re(a), Im(b)), Mul(Re(b), Im(a))), 0))))

c1bee1

\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, \log(z) = \log\!\left(-b\right) - \pi i

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Im}(a) < 0 \;\mathbin{\operatorname{and}}\; \operatorname{Im}(b) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) \operatorname{Im}(b) - \operatorname{Re}(b) \operatorname{Im}(a) < 0

TeX:

\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, \log(z) = \log\!\left(-b\right) - \pi i

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Im}(a) < 0 \;\mathbin{\operatorname{and}}\; \operatorname{Im}(b) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) \operatorname{Im}(b) - \operatorname{Re}(b) \operatorname{Im}(a) < 0

Definitions:

Fungrim symbol	Notation	Short description
`AnalyticContinuation`	$\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, f(z)$	Analytic continuation
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("c1bee1"),
    Formula(Equal(AnalyticContinuation(Log(z), For(z, a, b)), Sub(Log(Neg(b)), Mul(Pi, ConstI)))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC), Less(Im(a), 0), Greater(Im(b), 0), Less(Sub(Mul(Re(a), Im(b)), Mul(Re(b), Im(a))), 0))))

a2189a

\mathop{\text{Continuation}}\limits_{\displaystyle{z: \left(R {e}^{i t},\, t : 0 \rightsquigarrow \theta\right)}} \, \log(z) = \log(R) + \theta i

Assumptions:

R \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; \theta \in \mathbb{R}

TeX:

\mathop{\text{Continuation}}\limits_{\displaystyle{z: \left(R {e}^{i t},\, t : 0 \rightsquigarrow \theta\right)}} \, \log(z) = \log(R) + \theta i

R \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; \theta \in \mathbb{R}

Definitions:

Fungrim symbol	Notation	Short description
`AnalyticContinuation`	$\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, f(z)$	Analytic continuation
`Log`	$\log(z)$	Natural logarithm
`CurvePath`	$\left(f(t),\, t : a \rightsquigarrow b\right)$	Path along a curve
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("a2189a"),
    Formula(Equal(AnalyticContinuation(Log(z), For(z, CurvePath(Mul(R, Exp(Mul(ConstI, t))), For(t, 0, theta)))), Add(Log(R), Mul(theta, ConstI)))),
    Variables(R, theta),
    Assumptions(And(Element(R, OpenInterval(0, Infinity)), Element(theta, RR))))

cbfd70

\mathop{\text{Continuation}}\limits_{\displaystyle{t: 0 \rightsquigarrow \theta}} \, \log\!\left(R {e}^{i t}\right) = \log(R) + \theta i

Assumptions:

R \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; \theta \in \mathbb{R}

TeX:

\mathop{\text{Continuation}}\limits_{\displaystyle{t: 0 \rightsquigarrow \theta}} \, \log\!\left(R {e}^{i t}\right) = \log(R) + \theta i

R \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; \theta \in \mathbb{R}

Definitions:

Fungrim symbol	Notation	Short description
`AnalyticContinuation`	$\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, f(z)$	Analytic continuation
`Log`	$\log(z)$	Natural logarithm
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("cbfd70"),
    Formula(Equal(AnalyticContinuation(Log(Mul(R, Exp(Mul(ConstI, t)))), For(t, 0, theta)), Add(Log(R), Mul(theta, ConstI)))),
    Variables(R, theta),
    Assumptions(And(Element(R, OpenInterval(0, Infinity)), Element(theta, RR))))

1d447b

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \log(z) = \left\{1\right\}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \log(z) = \left\{1\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`Log`	$\log(z)$	Natural logarithm
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("1d447b"),
    Formula(Equal(Zeros(Log(z), ForElement(z, CC)), Set(1))))

13895b

\log\!\left(\overline{z}\right) = \overline{\log(z)}

Assumptions:

z \in \mathbb{C} \setminus \left(-\infty, 0\right]

TeX:

\log\!\left(\overline{z}\right) = \overline{\log(z)}

z \in \mathbb{C} \setminus \left(-\infty, 0\right]

Definitions:

Fungrim symbol	Notation	Short description
`Log`	$\log(z)$	Natural logarithm
`Conjugate`	$\overline{z}$	Complex conjugate
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("13895b"),
    Formula(Equal(Log(Conjugate(z)), Conjugate(Log(z)))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0)))))

099b19

\operatorname{Re}\!\left(\log(z)\right) = \log\!\left(\left|z\right|\right)

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

\operatorname{Re}\!\left(\log(z)\right) = \log\!\left(\left|z\right|\right)

z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Re`	$\operatorname{Re}(z)$	Real part
`Log`	$\log(z)$	Natural logarithm
`Abs`	$\left\|z\right\|$	Absolute value
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("099b19"),
    Formula(Equal(Re(Log(z)), Log(Abs(z)))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

fbfb81

\operatorname{Im}\!\left(\log(z)\right) = \arg(z)

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

\operatorname{Im}\!\left(\log(z)\right) = \arg(z)

z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Log`	$\log(z)$	Natural logarithm
`Arg`	$\arg(z)$	Complex argument
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("fbfb81"),
    Formula(Equal(Im(Log(z)), Arg(z))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

dcc1e5

\left|\log(z)\right| = \sqrt{\log^{2}\!\left(\left|z\right|\right) + {\left(\arg(z)\right)}^{2}}

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

\left|\log(z)\right| = \sqrt{\log^{2}\!\left(\left|z\right|\right) + {\left(\arg(z)\right)}^{2}}

z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Log`	$\log(z)$	Natural logarithm
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Arg`	$\arg(z)$	Complex argument
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("dcc1e5"),
    Formula(Equal(Abs(Log(z)), Sqrt(Add(Pow(Log(Abs(z)), 2), Pow(Arg(z), 2))))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

4986ed

\log(x) \le x - 1

Assumptions:

x \in \left(0, \infty\right)

TeX:

\log(x) \le x - 1

x \in \left(0, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`Log`	$\log(z)$	Natural logarithm
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("4986ed"),
    Formula(LessEqual(Log(x), Sub(x, 1))),
    Variables(x),
    Assumptions(Element(x, OpenInterval(0, Infinity))))

792c76

\left|\log(z)\right| \le \left|\log\!\left(\left|z\right|\right)\right| + \pi

Assumptions:

z \in \mathbb{C} \setminus \left\{0\right\}

TeX:

\left|\log(z)\right| \le \left|\log\!\left(\left|z\right|\right)\right| + \pi

z \in \mathbb{C} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("792c76"),
    Formula(LessEqual(Abs(Log(z)), Add(Abs(Log(Abs(z))), Pi))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, Set(0)))))

77aa12

\left|\log\!\left(x + a\right) - \log(x)\right| \le \log\!\left(1 + \frac{\left|a\right|}{x - \left|a\right|}\right)

Assumptions:

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; a \in \mathbb{R} \;\mathbin{\operatorname{and}}\; a \ge 0 \;\mathbin{\operatorname{and}}\; \left|a\right| < x

TeX:

\left|\log\!\left(x + a\right) - \log(x)\right| \le \log\!\left(1 + \frac{\left|a\right|}{x - \left|a\right|}\right)

x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; a \in \mathbb{R} \;\mathbin{\operatorname{and}}\; a \ge 0 \;\mathbin{\operatorname{and}}\; \left|a\right| < x

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Log`	$\log(z)$	Natural logarithm
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("77aa12"),
    Formula(LessEqual(Abs(Sub(Log(Add(x, a)), Log(x))), Log(Add(1, Div(Abs(a), Sub(x, Abs(a))))))),
    Variables(x, a),
    Assumptions(And(Element(x, RR), Element(a, RR), GreaterEqual(a, 0), Less(Abs(a), x))))

0ba9b2

\log(z) = \int_{1}^{z} \frac{1}{t} \, dt

Assumptions:

z \in \mathbb{C} \setminus \left(-\infty, 0\right]

TeX:

\log(z) = \int_{1}^{z} \frac{1}{t} \, dt

z \in \mathbb{C} \setminus \left(-\infty, 0\right]

Definitions:

Fungrim symbol	Notation	Short description
`Log`	$\log(z)$	Natural logarithm
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("0ba9b2"),
    Formula(Equal(Log(z), Integral(Div(1, t), For(t, 1, z)))),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0)))))

c77f9a

\int \frac{1}{z} \, dz = \log(z) + \mathcal{C}

Assumptions:

z \in \mathbb{C} \setminus \left(-\infty, 0\right]

TeX:

\int \frac{1}{z} \, dz = \log(z) + \mathcal{C}

z \in \mathbb{C} \setminus \left(-\infty, 0\right]

Definitions:

Fungrim symbol	Notation	Short description
`ComplexIndefiniteIntegralEqual`	$\int f(x) \, dx = g(x) + \mathcal{C}$	Indefinite integral, complex derivative
`Log`	$\log(z)$	Natural logarithm
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("c77f9a"),
    Formula(ComplexIndefiniteIntegralEqual(Div(1, z), Log(z), z)),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0)))))

e4f73a

\int \frac{1}{z} \, dz = \log\!\left(-z\right) + \mathcal{C}

Assumptions:

z \in \mathbb{C} \setminus \left[0, \infty\right)

TeX:

\int \frac{1}{z} \, dz = \log\!\left(-z\right) + \mathcal{C}

z \in \mathbb{C} \setminus \left[0, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`ComplexIndefiniteIntegralEqual`	$\int f(x) \, dx = g(x) + \mathcal{C}$	Indefinite integral, complex derivative
`Log`	$\log(z)$	Natural logarithm
`CC`	$\mathbb{C}$	Complex numbers
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("e4f73a"),
    Formula(ComplexIndefiniteIntegralEqual(Div(1, z), Log(Neg(z)), z)),
    Variables(z),
    Assumptions(Element(z, SetMinus(CC, ClosedOpenInterval(0, Infinity)))))

a4ac32

\int \frac{1}{x} \, dx = \log\!\left(\left|x\right|\right) + \mathcal{C}

Assumptions:

x \in \mathbb{R} \setminus \left\{0\right\}

TeX:

\int \frac{1}{x} \, dx = \log\!\left(\left|x\right|\right) + \mathcal{C}

x \in \mathbb{R} \setminus \left\{0\right\}

Definitions:

Fungrim symbol	Notation	Short description
`RealIndefiniteIntegralEqual`	$\int f(x) \, dx = g(x) + \mathcal{C}$	Indefinite integral, real derivative
`Log`	$\log(z)$	Natural logarithm
`Abs`	$\left\|z\right\|$	Absolute value
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("a4ac32"),
    Formula(RealIndefiniteIntegralEqual(Div(1, x), Log(Abs(x)), x)),
    Variables(x),
    Assumptions(Element(x, SetMinus(RR, Set(0)))))

Definitions

Illustrations

Particular values

Functional equations and connection formulas

Analytic properties

Complex parts

Bounds and inequalities

Integral representations