# Limit of a complex function

Can someone show me how to find the limit of a complex function?

Example:

z1 = 3 + 4*I

some_function[z] = z * z1

Set-up:

Limit[some_function[z],z->z1]

The result should be

(4*I + 3)*z

but I get

Limit[some_function[z], z -> 3 + 4 I]

What am I doing wrong? I tried to search in google and in the help files in the section about complex analysis.

Thank you for your attention.

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## 2 Answers

can't use _ in names. the _ is used for patterns. Also your function definitions are not correct. Need to use delay assignment :=

May be this will work for you:

ClearAll[z, z1];
z1 = 3 + 4*I;
someFunction[z_] := z*z1;
Limit[someFunction[z], z -> z1]

(*-7 + 24 I*)
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Thank you, @Nasser – Luiz Roberto Meier Nov 17 '13 at 10:57

Taking a sample complex function as example with z->1+i

eq = z^2 - 2 z + 1;

Separating complex and Real part,

eqt = ComplexExpand[eq //. {z -> x + I y}]

1 - 2 x + x^2 - y^2 + I (-2 y + 2 x y)

Taking real part,

Cases[eqt, x_ /; FreeQ[x, I]] /. List -> Plus

1 - 2 x + x^2 - y^2 Calculating limit for real part as,

Limit[Limit[1 - 2 x + x^2 - y^2, x -> 1], y -> 1]

-1

Taking Complex part,

Complement[eqt, 1 - 2 x + x^2 - y^2]

I (-2 y + 2 x y) Taking limit as,

Limit[Limit[(-2 y + 2 x y), x -> 1], y -> 1]

0

So, by theorems of Complex Analysis its limit is -1. Use Example 2.17 on this Page

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