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`Complex Function Complex Exponential Function Properties`

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# Complex Function

## Complex Exponential Function

For the function, 𝑓(𝑧)=ℯ𝑥cos𝑦+𝑖ℯ𝑥sin𝑦, (where 𝑧=𝑥+𝑖𝑦) is an entire (=analytic in function.

Some of its properties:

• if 𝑦=0, then 𝑓(𝑧)=𝑓(𝑥+𝑖⋅0)=𝑓(𝑥)=ℯ𝑥, so 𝑓 agrees with the "regular" exponential function on ℝ
• 𝑓(𝑧)=ℯ𝑥(cos𝑦+𝑖sin𝑦)=ℯ𝑥𝑖𝑦

By definition. The complex exponential function, 𝑧, sometimes also denoted exp(𝑧), is defined by

`ℯ𝑧=ℯ𝑥⋅ℯ𝑖𝑦, where 𝑧=𝑥+𝑖𝑦`

## Properties

For the function, 𝑧= ℯ𝑥⋅ℯ𝑖𝑦, where 𝑧=𝑥+𝑖𝑦:

• |ℯ𝑧|=|ℯ𝑥||ℯ𝑖𝑦|=ℯ𝑥
• arg𝑧=arg(ℯ𝑥𝑖𝑦)=𝑦(+2𝜋𝑘, where 𝑘∈ℤ)
• 𝑧+2𝜋𝑖=ℯ𝑥𝑖(𝑦+2𝜋)=ℯ𝑥𝑖𝑦=ℯ𝑧
• ```ℯ𝑧+𝑤=ℯ(𝑥+𝑖𝑦)+(𝑢+𝑖𝑣), where 𝑧=𝑥+𝑖𝑦, 𝑤=𝑢+𝑖𝑣  =ℯ(𝑥+𝑢)+𝑖(𝑦+𝑣)=ℯ𝑥ℯ𝑢ℯ𝑖𝑦ℯ𝑖𝑦  =(ℯ𝑥ℯ𝑖𝑦)(ℯ𝑢ℯ𝑖𝑦)=ℯ𝑧ℯ𝑤```
• 1𝑧=ℯ−𝑧, since ℯ𝑧−𝑧=ℯ0=1
• 𝑧 is an entire function.
• Derivative 𝑓′(𝑧):

Let 𝑢(𝑥,𝑦)=ℯ𝑥cos𝑦, 𝑣(𝑥,𝑦)=ℯ𝑥sin𝑦

Then ```𝑢𝑥(𝑥,𝑦)=𝑒𝑥cos𝑦;𝑣𝑥(𝑥,𝑦)=𝑒𝑥sin𝑦 𝑢𝑦(𝑥,𝑦)=−𝑒𝑥sin𝑦;𝑣𝑦(𝑥,𝑦)=𝑒𝑥cos𝑦```

Thus 𝑓′(𝑧)=𝑢(𝑥,𝑦)+𝑖𝑣(𝑥,𝑦)=ℯ𝑥cos𝑦+𝑖ℯ𝑥sin𝑦=ℯ𝑧

So the derivative of 𝑧 is 𝑧, in symbols, `dd𝑧ℯ𝑧=ℯ𝑧`.

• `dd𝑧ℯ𝑎𝑧=𝑎⋅ℯ𝑎𝑧 (𝑎∈ℂ)` by the chain rule
• 𝑧=ℯ𝑥−𝑖𝑦=ℯ𝑥−𝑖𝑦=ℯ𝑥𝑖𝑦=𝑥𝑖𝑦=𝑧
• 𝑧=1 if and only if 𝑥𝑖𝑦=1. The complex number in polar form, 𝑥𝑖𝑦, equals 1, when its length equals 1 and its argument equals 0, ie.e. when 𝑥 and y=2𝑘𝜋, 𝑘∈ℤ. Thus

`ℯ𝑧=1⇔𝑧=2𝜋𝑖𝑘, 𝑘∈ℤ`
• 𝑧=ℯ𝑤⇔ℯ𝑧−𝑤=1⇔𝑧−𝑤=2𝜋𝑖𝑘⇔𝑧=𝑤+2𝜋𝑖𝑘

The function 𝑤=ℯ𝑧 is a mapping from `ℂ 𝑧-plane ` to `ℂ 𝑤-plane `.

For the images of horizontal lines, 𝐿={𝑥+𝑖𝑦0|𝑥∈ℝ} for fixed 𝑦0∈ℝ. Then 𝑧=ℯ𝑥+𝑖𝑦0=ℯ𝑥𝑖𝑦0, a line from origin but not equal with fixed angle.

For the images of vertical lines, 𝐿={𝑥0+𝑖𝑦|𝑦∈ℝ} for fixed 𝑥0∈ℝ. Then 𝑧=ℯ𝑥0+𝑖𝑦=ℯ𝑥0𝑖𝑦, a circle with center at origin.

For the images of vertical strip, 𝑆={𝑧:0<Re𝑧<1}, a ring between circle of value 0 and e

• When 𝑧=0
```ℯ𝑧=0⇔ℯ𝑥⋅ℯ𝑖𝑦=0 Note: ℯ𝑖𝑦 has absolute value 1  ⇔ℯ𝑥=0  ⇔Never...!```
• For a given 𝑧∈ℂ\{0}, is there a 𝑤∈ℂ such that 𝑤=𝑧? Writing 𝑧=|𝑧|ℯ𝑖𝜃 and 𝑤=𝑢+𝑖𝑣 this is equivalent to:
```ℯ𝑤=𝑧⇔ℯ𝑢ℯ𝑖𝑣=|𝑧|ℯ𝑖𝜃  ⇔ℯ𝑢=|𝑧| and ℯ𝑖𝑣=ℯ𝑖𝜃  ⇔𝑢=ln|𝑧| and 𝑣=𝜃+2𝑘𝜋  ⇔𝑤=ln|𝑧|+𝑖arg𝑧```

This is the complex logarithm.

ID: 190400003 Last Updated: 3/4/2019 Revision: 0 Home 5

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