page_title: Complex logarithm
hierarchical_section_title: Complex logarithm
caption_attribution_description: function Log[z] in the complex plane
context_page_description: In complex analysis, a complex logarithm of the non-zero complex number z, denoted by w = log z, is defined to be any complex number w for which e ʷ = z. This construction is analogous to the real logarithm function ln, which is the inverse of the real exponential function e ʸ, satisfying e ˡⁿ ˣ = x for positive real numbers x.
  If a non-zero complex number z is given in polar form as z = re iθ, then w₀ = ln + iθ is one logarithm of z. Since z = re exactly for all integer k, adding integer multiples of 2π to the argument θ gives all the numbers that are logarithms of z:
  wₖ = ln + i.
  All these complex logarithms of z are on a vertical line in the complex plane with real part ln.
  Since any nonzero complex number has infinitely many complex logarithms, the complex logarithm cannot be defined to be a single-valued function on the complex numbers, but only as a multivalued function. Settings for a formal treatment of this are, among others, the associated Riemann surface, branches, or partial inverses of the complex exponential function.
  Sometimes the notation ln instead of log is used when addressing the complex logarithm.
context_section_description: In complex analysis, a complex logarithm of the non-zero complex number z, denoted by w = log z, is defined to be any complex number w for which e ʷ = z. This construction is analogous to the real logarithm function ln, which is the inverse  of the real exponential function e ʸ, satisfying e ˡⁿ ˣ = x for positive real numbers x.
  If a non-zero complex number z is given in polar form as z = re iθ (r and θ real numbers, with r > 0), then w₀ = ln(r) + iθ is one logarithm of z. Since z = re exactly for all integer k, adding integer multiples of 2π to the argument θ gives all the numbers that are logarithms of z:
  wₖ = ln(r) + i(θ + 2kπ).
  All these complex logarithms of z are on a vertical line in the complex plane with real part ln(r).
  Since any nonzero complex number has infinitely many complex logarithms, the complex logarithm cannot be defined to be a single-valued function on the complex numbers, but only as a multivalued function. Settings for a formal treatment of this are, among others, the associated Riemann surface, branches, or partial inverses of the complex exponential function.
  Sometimes the notation ln instead of log is used when addressing the complex logarithm.
caption: A single branch of the complex logarithm. The hue of the color is used to show the arg (polar coordinate angle) of the complex logarithm. The saturation and value (intensity and brightness) of the color is used to show the modulus of the complex logarithm.
exif: {"Image Orientation": "Horizontal (normal)", "Image XResolution": "28", "Image YResolution": "28", "Image ResolutionUnit": "Pixels/Centimeter", "Image Software": "Adobe Photoshop 7.0", "Image DateTime": "2007:08:01 17:53:13", "Image ExifOffset": "156", "Thumbnail Compression": "JPEG (old-style)", "Thumbnail XResolution": "72", "Thumbnail YResolution": "72", "Thumbnail ResolutionUnit": "Pixels/Inch", "Thumbnail JPEGInterchangeFormat": "294", "Thumbnail JPEGInterchangeFormatLength": "2701", "EXIF ColorSpace": "Uncalibrated", "EXIF ExifImageWidth": "636", "EXIF ExifImageLength": "636"}