Proposal for gradient wrt complex variables

This is a proposal to handle gradients of a scalar, real variable (usually, a cost) with respect to tensor variables, of complex (and real) type, in an optimization perspective.

Derivative of complex variables is usually studied only for so-called analytical complex functions, which have a particular structure in their partial derivatives. However, we do not want to limit ourselves to analytical functions, and we make other assumptions (that the final cost is real-valued, for instance), so we will adopt a different convention for gradients than what is usually used in the literature.

Gradient (re-)definition

We are interested in the case where we have a final real-valued cost,

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, and a graph of mathematical expressions, including real-valued and complex-valued variables (scalars, vectors, matrices, higher-order tensors), and we want to compute the gradient of

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, wrt some variables in that graph, using gradient back-propagation. In the case where some variables are complex, the usual chain rule cannot be applied, except in some cases.

For each real-valued variable

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(not necessarily scalar, it could be a matrix, for instance), in particular

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and

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, partial derivatives can be defined:

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has the same number of dimensions and shape as

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. We will limit that notation to real-valued variables only, this way, the partial derivative itself will be real-valued too. We will not use that notation for the complex derivative of analytical complex functions.

For any real-valued intermediate variable

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, the usual chain rule applies:

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If

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is a complex variable, with

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and

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, we can consider

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and

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as free variables, and then:

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If we want to use an algorithm similar to gradient backpropagation, we can see that, here, we need to have both

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and

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, in order to compute

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.

For each variable

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in the expression graph, let us denote

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the gradient of

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with respect to

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. It is a tensor with the same dimensions as

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, and can be complex-valued. We define:

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This is the tensor that we are going to back-propagate through the computation graph.

Generalized chain rule

Using the definition above, if we have two complex variables

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and

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(with

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all real-valued):

System Message: WARNING/2 (\nabla_C(z) &= \frac{\partial C}{\partial \Re z} + i \frac{\partial C}{\partial \Im z} \\ &= \frac{\partial C}{\partial x} + i \frac{\partial C}{\partial y} \nabla_C(t) &= \frac{\partial C}{\partial \Re t} + i \frac{\partial C}{\partial \Im t} \\ &= \frac{\partial C}{\partial r} + i \frac{\partial C}{\partial s} \\ &= \left(\frac{\partial C}{\partial x} \frac{\partial x}{\partial r} + \frac{\partial C}{\partial y} \frac{\partial y}{\partial r}\right) + i \left(\frac{\partial C}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial C}{\partial y} \frac{\partial y}{\partial s}\right) \\ &= \frac{\partial C}{\partial x} \left(\frac{\partial x}{\partial r} + i \frac{\partial x}{\partial s}\right) + \frac{\partial C}{\partial y} \left(\frac{\partial y}{\partial r} + i \frac{\partial y}{\partial s}\right) \\ &= \Re \left(\nabla_C(z)\right) \left(\frac{\partial x}{\partial r} + i \frac{\partial x}{\partial s}\right) + \Im \left(\nabla_C(z)\right) \left(\frac{\partial y}{\partial r} + i \frac{\partial y}{\partial s}\right))

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This formula can be used whether or not

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is an analytical function of

System Message: WARNING/2 (z)

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or

System Message: WARNING/2 (t)

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.
, and whether or not

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is an analytical function of

System Message: WARNING/2 (t)

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.
.

Special cases

Real-valued input variable

If variable

System Message: WARNING/2 (x)

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is defined as real-valued, it can sometimes be useful to have the value of

System Message: WARNING/2 (\nabla_C(z))

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.
instead of only

System Message: WARNING/2 (\frac{\partial C}{\partial x})

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, because the imaginary part contains information on how the cost would change if

System Message: WARNING/2 (y)

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.
was not constrained to be 0.

Real-valued intermediate variable

When

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is an intermediate variable, however, the gradient of

System Message: WARNING/2 (C)

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.
wrt

System Message: WARNING/2 (t)

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must not be backpropagated through

System Message: WARNING/2 (y)

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. Therefore, we have:

System Message: WARNING/2 (\nabla_C(t) &= \frac{\partial C}{\partial r} + i \frac{\partial C}{\partial s} \\ &= \frac{\partial C}{\partial x} \frac{\partial x}{\partial r} + i \frac{\partial C}{\partial x} \frac{\partial x}{\partial s} \\ &= \Re \left(\nabla_C(z)\right) \left(\frac{\partial x}{\partial r} + i \frac{\partial x}{\partial s}\right))

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.

The imaginary part of

System Message: WARNING/2 (\nabla_C(z))

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is ignored, because

System Message: WARNING/2 (\Im z)

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is constrained to be 0.

Analytic functions

If

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is the output of an analytic function of

System Message: WARNING/2 (t)

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.
, some simplifications are possible. Analytic functions include, for instance, polynomial functions, the exponential function. Most complex functions, however, are not: absolute value, real part, imaginary part, complex conjugate, etc.

Analytic (or holomorphic) functions satisfy the Cauchy-Riemann equations:

System Message: WARNING/2 (\frac{\partial \Re z}{\partial \Re t} = \frac{\partial \Im z}{\partial \Im t} \text{ and } \frac{\partial \Re z}{\partial \Im t} = - \frac{\partial \Im z}{\partial \Re t})

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.

Or, in our case:

System Message: WARNING/2 (\frac{\partial x}{\partial r} = \frac{\partial y}{\partial t} \text{ and } \frac{\partial x}{\partial s} = - \frac{\partial y}{\partial r})

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.

This leads to:

System Message: WARNING/2 (\nabla_C(t) &= \Re \left(\nabla_C(z)\right) \left(\frac{\partial x}{\partial r} + i \frac{\partial x}{\partial s}\right) + \Im \left(\nabla_C(z)\right) \left(\frac{\partial y}{\partial r} + i \frac{\partial y}{\partial s}\right) \\ &= \Re \left(\nabla_C(z)\right) \left(\frac{\partial x}{\partial r} + i \frac{\partial x}{\partial s}\right) + \Im \left(\nabla_C(z)\right) \left(- \frac{\partial x}{\partial s} + i \frac{\partial x}{\partial r}\right) \\ &= \Re \left(\nabla_C(z)\right) \left(\frac{\partial x}{\partial r} + i \frac{\partial x}{\partial s}\right) + i \Im \left(\nabla_C(z)\right) \left(\frac{\partial x}{\partial r} + i \frac{\partial x}{\partial s}\right) \\ \nabla_C(t) &= \nabla_C(z) \left(\frac{\partial x}{\partial r} + i \frac{\partial x}{\partial s}\right) = - i \nabla_C(z) \left(\frac{\partial y}{\partial r} + i \frac{\partial y}{\partial s}\right))

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.

Finite differences

In order to verify that the mathematical formula for a gradient, or its implementation, is correct, we usually use a finite-differenciation approach. If

System Message: WARNING/2 (C)

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is our real scalar cost, and

System Message: WARNING/2 (x)

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.
a real-valued scalar variable, then:

System Message: WARNING/2 (\frac{\partial C}{\partial x} \approx \frac{C(x + \varepsilon) - C(x)}{\varepsilon})

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.

where

System Message: WARNING/2 (\varepsilon)

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is also a real scalar, of small magnitude (typically

System Message: WARNING/2 (10^{-6})

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.
to

System Message: WARNING/2 (10^{-4})

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). If

System Message: WARNING/2 (x)

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is a tensor, then this approximation has to be made for each element

System Message: WARNING/2 (x_i)

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.
independently (a different

System Message: WARNING/2 (\varepsilon_i)

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could be used each time, but usually they are all equal to

System Message: WARNING/2 (\varepsilon)

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.
).

For a complex scalar variable

System Message: WARNING/2 (z = x + iy)

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.
:

System Message: WARNING/2 (\nabla_C(z) &= \frac{\partial C}{\partial x} + i \frac{\partial C}{\partial y}\\ \nabla_C(z) &\approx \frac{C(z + \delta) - C(z)}{\delta} + i \frac{C(z + i \varepsilon) - C(z)}{\varepsilon})

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.

Both partial derivative have to be estimated independently, using generally

System Message: WARNING/2 (\delta = \varepsilon)

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.
.