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Complex Function

Complex Exponential Function

For the function, 𝑓(𝑧)=ℯ^𝑥cos𝑦+𝑖ℯ^𝑥sin𝑦, (where 𝑧=𝑥+𝑖𝑦) is an entire (=analytic in ℂ function.

Some of its properties:

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

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

ℯ𝑧=ℯ𝑥⋅ℯ𝑖𝑦, where 𝑧=𝑥+𝑖𝑦

Properties

For the function, ℯ^𝑧= ℯ^𝑥⋅ℯ^𝑖𝑦, where 𝑧=𝑥+𝑖𝑦:

• |ℯ^𝑧|=|ℯ^𝑥||ℯ^𝑖𝑦|=ℯ^𝑥
• argℯ^𝑧=arg(ℯ^𝑥ℯ^𝑖𝑦)=𝑦(+2𝜋𝑘, where 𝑘∈ℤ)
• ℯ^{𝑧+2𝜋𝑖}=ℯ^𝑥ℯ^{𝑖(𝑦+2𝜋)}=ℯ^𝑥ℯ^𝑖𝑦=ℯ^𝑧
• ℯ𝑧+𝑤=ℯ(𝑥+𝑖𝑦)+(𝑢+𝑖𝑣), where 𝑧=𝑥+𝑖𝑦, 𝑤=𝑢+𝑖𝑣  =ℯ(𝑥+𝑢)+𝑖(𝑦+𝑣)=ℯ𝑥ℯ𝑢ℯ𝑖𝑦ℯ𝑖𝑦  =(ℯ𝑥ℯ𝑖𝑦)(ℯ𝑢ℯ𝑖𝑦)=ℯ𝑧ℯ𝑤
• 1ℯ^𝑧=ℯ^−𝑧, since ℯ^𝑧ℯ^−𝑧=ℯ⁰=1
• ℯ^𝑧 is an entire function.
• Derivative 𝑓′(𝑧):

  Let 𝑢(𝑥,𝑦)=ℯ^𝑥cos𝑦, 𝑣(𝑥,𝑦)=ℯ^𝑥sin𝑦

  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)=ℯ^𝑥cos𝑦+𝑖ℯ^𝑥sin𝑦=ℯ^𝑧

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

• dd𝑧ℯ𝑎𝑧=𝑎⋅ℯ𝑎𝑧 (𝑎∈ℂ) by the chain rule
• ℯ^𝑧=ℯ^{𝑥−𝑖𝑦}=ℯ^𝑥ℯ^−𝑖𝑦=ℯ^𝑥ℯ^𝑖𝑦=ℯ^𝑥ℯ^𝑖𝑦=ℯ^𝑧

• ℯ^𝑧=1 if and only if ℯ^𝑥ℯ^𝑖𝑦=1. The complex number in polar form, ℯ^𝑥ℯ^𝑖𝑦, equals 1, when its length equals 1 and its argument equals 0, ie.e. when ℯ^𝑥 and y=2𝑘𝜋, 𝑘∈ℤ. Thus

  ℯ𝑧=1⇔𝑧=2𝜋𝑖𝑘, 𝑘∈ℤ
• ℯ^𝑧=ℯ^𝑤⇔ℯ^𝑧−𝑤=1⇔𝑧−𝑤=2𝜋𝑖𝑘⇔𝑧=𝑤+2𝜋𝑖𝑘

  The function 𝑤=ℯ^𝑧 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 ℯ^𝑧=ℯ^{𝑥+𝑖𝑦₀}=ℯ^𝑥ℯ^𝑖𝑦₀, 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 ℯ^𝑧=ℯ^{𝑥₀+𝑖𝑦}=ℯ^𝑥₀ℯ^𝑖𝑦, 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 ℯ^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 ℯ^𝑤=𝑧? Writing 𝑧=|𝑧|ℯ^𝑖𝜃 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.