**Relationship between real and Complex differentiability** The most commonly used definition of differentiability of a function is as follows. *The function is (real) differentiable at if the limit*

*exists*. Equivalently, We say that is differentiable at if there exists a linear transformation satisfying

Such a linear transformation, if it exists, is unique and is called the derivative of at . We then write . More generally, a function is differentiable at if there exists a linear transformation such that

where the norms and are the standard norms on and respectively. The linear transformation, if it exists, is unique and is called the *Jacobian* of at and is denoted by . If we write and , then we can write the Jacobian in matrix form as

Before looking at complex differentiability we recall **Proposition:** Let be-linear. The following statements are equivalent:

- There exists suchthat ;
- is -linear;
- ;
- The matrix with respect to the canonical basis has the form

Now, a function is *complex-differentiable* at if the limit

exists. This can also be reformulated as follows. We say that is complex-differentiable at if there exists such that

and we have .

From the above discussion, it is clear that if is complex-differentiable at , then it is real-differentiable at and the Jacobian is -linear.

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August 19, 2011 at 8:37 PM

Steven SpalloneThey recently figured out the quaternionic case too: see recent work by

Graziano Gentili and Fabio Vlacci!