with lines and lines in the complex -plane.

complex logarithm drawn with a color map

In complex analysis, the complex logarithm is the extension of the natural logarithm function ln(x) – originally defined for real numbers only – into the complex plane. Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

The natural logarithm can be defined for all non-zero complex numbers z. It is usually denoted log(z), in line with the notational conventions of complex analysis, where the variable z is commonly used to denote a complex variable, just as x customarily stands for a real variable. The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

The complex logarithm is given in terms of the polar representation of a complex number z=re^iθ by

The logarithm is well-defined by this expression, but for an ambiguity in the angle θ.

Derivation from the Taylor series

From the viewpoint of analysis, the most natural way to define log(z) is by the method of analytic continuation, from the positive real number line (where log(x) = ln(x)) into the complex plane. Here's how that works. Analysis has its beginnings in the rigorous formulation of calculus. ... In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

The Taylor series (also known as the Mercator series) Series expansion redirects here. ... In mathematics, the Mercator series or Newton-Mercator series is the Taylor series for the natural logarithm. ...

converges uniformly for all complex numbers z with absolute value less than 1. And by applying Abel's test to the Taylor series for log(1−z), it can be shown that the series for log(1+z) converges when |z| = 1, except when z = −1 (in other words, log(0) does not exist). In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... In mathematics, Abels test is a method of testing for the convergence of a series and is named after mathematician Neils Abel. ...

A careful analysis of the power series for In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...

shows that this series converges for |z| ≤ 1, except when z = ±1.^[1] By plugging

into the power series direct calculation reveals that

where the last equality follows from the well-known series for the (real) inverse tangent function.

By supposing that the laws of (real) logarithms apply to complex logarithms we obtain two results:^[2]

The fact that log(−1) has two values can be explained this way. The complex logarithm is only continuous in a finite neighborhood of a point z, and log(0) does not exist. The "path" from z = 1 through z = i to z = −1 leads, by continuity, to one value of log(−1), while the "path" that passes through z = −i leads, by continuity, to the other value. The two paths lead to two different values because they encircle a branch point at z = 0. This multi-valued aspect of the complex logarithm is traditionally handled by making a "cut" in the complex plane all along the negative real axis. And the preceding explanation is, historically, why the argument of a complex number is often specified in the range from −π to π.^[3] In complex analysis, a branch point may be thought of informally as a point z0 at which a multiple_valued function changes values when one winds once around z0. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ...

Log(z) as the inverse of the exponential function

If the non-zero complex number z is expressed in polar coordinates as z = re^iθ with r > 0 and −π < θ ≤ π, then A polar grid with several angles labeled in degrees In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by an angle and a distance. ...

where ln(r) is the usual natural logarithm of a real number.

So defined, log (z) is holomorphic for all complex numbers which are not real numbers ≤ 0, and it has the property Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...

for all nonzero z. Some familiar properties of the real-valued natural logarithm are no longer valid for this complex extension. For example, log(e^z) does not always equal z, and log(zw) does not always equal log(z) + log(w) – in either case, the result might have to be adjusted modulo 2πi to stay within the range of this principal branch of the complex log function. The word modulo (Latin, with respect to a modulus of ___) is the Latin ablative of modulus which itself means a small measure. ... In mathematics, a principal branch is a function which selects one branch, or slice, of a multi-valued function. ...

Log(z) as a multi-valued function

A visualization of the Riemann surface of log(z). The surface appears to spiral around a vertical line corresponding to the origin of the complex plane. The actual surface extends arbitrarily far outward, but is cut off in the visualization.

A somewhat more natural definition of log(z) interprets it as a multi-valued function, or, more precisely, as a multi-valued relationship.^[4] For z = re^iθ, it would be possible to choose This diagram does not represent a true function; because the element 3, in X, is associated with two elements b and c, in Y. In mathematics, a multivalued function is a total relation; i. ...

for any integer k. There is no other complex number u for which e^u =  z, because the period of e^z is 2πi (see Euler's identity). Not to be confused with Natural number. ... For other uses, see List of topics named after Leonhard Euler. ...

The standard way to deal with multi-valued relationships such as this in complex analysis is via Riemann surfaces: the function log(z) is then defined not on the complex plane, but on a suitable Riemann surface having countably many "leaves" or "sheets" instead. The function is then single-valued on each sheet, and the values of the function differ by 2πi from one sheet to the next. Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ...

More precisely, the logarithm has a branch point at z = 0 and another at the point at infinity, where the point at infinity is understood to be the point that compactifies the complex plane into the Riemann sphere. There is a branch cut between these two points; by convention, it is placed so that the cut runs off to the left, along the negative real axis, between z = 0 and z = −∞. The location of this cut is by convention only: it could be placed running from the origin in any direction, and some textbooks show the cut running downwards. In complex analysis, a branch point may be thought of informally as a point z0 at which a multiple_valued function changes values when one winds once around z0. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... The Riemann sphere can be visualized as the complex number plane wrapped around a sphere (by some form of stereographic projection — details are given below). ... In complex analysis, a branch point may be thought of informally as a point z0 at which a multiple_valued function changes values when one winds once around z0. ...

At the cut, different sheets join together, and a continuous path moving through the cut simply goes from one sheet to the next. The glueing of the sheets is such that the phase differs by 2π between two sheets at any point that is not on the cut, and by 0 at the cut itself. On the Riemann surface, then, log(−x) for positive, real x, can be defined uniquely, as long as it is made clear which sheet one is on; otherwise, by convention, log(−x) lies in the branch cut and remains undefined.

The complex logarithm as a conformal mapping

Since the derivative of log(z), z⁻¹, is a meromorphic function with one simple pole at z = 0, the principal branch of log(z) is a conformal map from ℂ/{0} into an infinitely long strip of height 2π, −π < Im(w) ≤ π. We can gain insight into the way this mapping works by writing it as In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function. ... In complex analysis, a pole of a function is a certain type of simple singularity that behaves like the singularity of f(z) = 1/zn at z = 0; a pole of a function f is a point a such that f(z) approaches infinity as z approaches a. ... In mathematics, a conformal map is a function which preserves angles. ...

and then analyzing the images in the w-plane of certain circles and straight lines in the z-plane.^[5]

• The unit circle in the z-plane is mapped into a segment of the imaginary axis in the w-plane, running from −πi to πi. Notice that the length of the resulting straight line segment equals the circumference of the original circle.

• The interior of the unit circle is mapped into the left-hand portion of the strip; that is, into the region where Re(w) < 0. More precisely, each circle centered on the origin and of radius r < 1, |z| = r, is mapped into the straight line segment Re(w) = ln(|z|). Each of these line segments is longer than the circle whose image it is, and the smaller the circle, the more it gets stretched out.

• The exterior of the unit circle is mapped into the right-hand portion of the strip; each circle |z| = r > 1 is straightened out and compressed to fit on the line segment Re(w) = ln(|z|).

• The positive real axis in the z-plane is mapped into the entire real axis in the w-plane. More precisely, if z₀ is real and z₀ > 0, its image w₀ is given by w₀ = ln(z₀). The negative real axis is mapped into the straight line Im(w) = π, with one notable difference – its direction is reversed. That is to say, the negative real axis, which runs from right to left in the z-plane, is mapped into a straight line running from left to right in the w-plane.

• All the rays originating at z = 0 are mapped into parallel horizontal lines in the w-plane. Rays that are "close" to the positive real axis lie close together in the w-plane, and rays that are '"close" to the negative real axis are farther apart, because the image of each ray arg(z) = θ is the straight line Im(w) = θ.

Since the mapping w = log(z) transforms circles into vertical straight line segments, it is useful in engineering applications involving an annulus, or doughnut-shaped region; the image of the annulus is a rectangle of height 2π, whose width depends on the thickness of the annulus. By the same token, the inverse mapping z = e^w transforms a rectangle into an annulus. Illustration of a unit circle. ... An annulus In mathematics, an annulus (the Latin word for little ring, with plural annuli) is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. ...

Log(z) also serves as a particularly simple illustration of the principle that conformal maps preserve angles. All the circles and radial lines discussed above are mutually perpendicular, as are the vertical and horizontal lines into which they are transformed.

Finally, the complete map of the corkscrew Riemann surface on which log(z) is holomorphic into the complex plane is easily visualized – each "sheet" of that surface is mapped into an infinitely wide strip of height 2π, and those strips cover the entire complex plane like wallpaper.

Imaginary-base logarithms

An imaginary-base logarithm is a logarithm with base a real multiple of i. These logarithms can be calculated with the formula Look up logarithm in Wiktionary, the free dictionary. ...

Where z is any complex number but 0.

Proof of the formula

Using Euler's formula, we know that Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...

Taking the natural logarithm of each side yields The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ...

Changing the base: Look up logarithm in Wiktionary, the free dictionary. ...

Multiples of i for bases

If the base is a multiple of i, the formula can be generalized as follows:

Where k is any real number.

This can be proved as follows:

Plots of the natural logarithm function on the complex plane (principal branch)

See also

• Complex argument (continued fraction)
• Inverse trigonometric function
• Logarithm

In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. ... Look up logarithm in Wiktionary, the free dictionary. ...

Notes

1. ^ The point z = 0 and the point at infinity are branch points for the complex logarithm function precisely because this series does not converge when z = ±1.
2. ^ These two results can also be obtained by expanding log(z) as a Taylor series about the points z = ±i, and then evaluating log(−1) with those two infinite series.
3. ^ (Whittaker and Watson, 1927, Appendix).
4. ^ A mathematical function is defined to be a single-valued relationship, so that a multi-valued function is a contradiction in terms, strictly speaking.
5. ^ This mapping can also be studied in conjunction with the inverse mapping w = e^z. See (Moretti, 1964, pp. 351-355), for example.

In complex analysis, a branch point may be thought of informally as a point z0 at which a multiple_valued function changes values when one winds once around z0. ...

References

• Gino Moretti, Functions of a Complex Variable, Prentice-Hall, Inc., 1964.
• E. T. Whittaker and G. N. Watson, A Course in Modern Analysis, fourth edition, Cambridge University Press, 1927.