source/reference:
https://www.youtube.com/channel/UCaTLkDn9_1Wy5TRYfVULYUw/playlists

Complex Function

Complex Trigonometric Function

For the function, ℯ^𝑖𝜃=cos𝜃+𝑖sin𝜃

Let ℯ𝑖𝜃=cos𝜃+𝑖sin𝜃 Therefore, ℯ−𝑖𝜃=cos(−𝜃)+𝑖sin(−𝜃)=cos𝜃−𝑖sin𝜃 Hence, ℯ𝑖𝜃+ℯ−𝑖𝜃=2cos𝜃 and ℯ𝑖𝜃−ℯ−𝑖𝜃=2𝑖sin𝜃 Thus cos𝜃=ℯ𝑖𝜃+ℯ−𝑖𝜃2 and sin𝜃=ℯ𝑖𝜃−ℯ−𝑖𝜃2𝑖

By definition, The complex cosine and sine functions are defined via

cos𝑧=ℯ𝑖𝑧+ℯ−𝑖𝑧2 and sin𝑧=ℯ𝑖𝑧−ℯ−𝑖𝑧2𝑖

Properties of Sine and Cosine

For the function, cos𝑧=ℯ^𝑖𝑧+ℯ^−𝑖𝑧2 and sin𝑧=ℯ^𝑖𝑧−ℯ^−𝑖𝑧2𝑖

• cos𝑧 and sin𝑧 are analytic functions (in fact, entire).
• For real-valued 𝑧, (i.e. 𝑧=𝑥+𝑖⋅0) the complex sine and cosine agree with the real-valued sine and cosine functions.
• cos(−𝑧)=ℯ^−𝑖𝑧+ℯ^𝑖𝑧2=cos𝑧
• sin(−𝑧)=ℯ^−𝑖𝑧−ℯ^𝑖𝑧2𝑖=−sin𝑧
• cos(𝑧+𝑤)=cos𝑧cos𝑤−sin𝑧sin𝑤, sin(𝑧+𝑤)=sin𝑧cos𝑤+cos𝑧sin𝑤

  Proofs of the addition formulae cos(𝑧+𝑤)

   cos𝑧cos𝑤−sin𝑧sin𝑤 =(ℯ𝑖𝑧+ℯ−𝑖𝑧2)(ℯ𝑖𝑤+ℯ−𝑖𝑤2) −(ℯ𝑖𝑧−ℯ−𝑖𝑧2𝑖)(ℯ𝑖𝑤−ℯ−𝑖𝑤2𝑖) =(ℯ𝑖𝑧+ℯ−𝑖𝑧)(ℯ𝑖𝑤+ℯ−𝑖𝑤)+(ℯ𝑖𝑧−ℯ−𝑖𝑧)(ℯ𝑖𝑤−ℯ−𝑖𝑤)4 =ℯ𝑖𝑧ℯ𝑖𝑤+ℯ𝑖𝑧ℯ−𝑖𝑤+ℯ−𝑖𝑧ℯ𝑖𝑤+ℯ−𝑖𝑧ℯ−𝑖𝑤+ℯ𝑖𝑧ℯ𝑖𝑤−ℯ𝑖𝑧ℯ−𝑖𝑤−ℯ−𝑖𝑧ℯ𝑖𝑤+ℯ−𝑖𝑧ℯ−𝑖𝑤4 =2ℯ𝑖𝑧ℯ𝑖𝑤+2ℯ−𝑖𝑧ℯ−𝑖𝑤4 =ℯ𝑖(𝑧+𝑤)+ℯ−𝑖(𝑧+𝑤)2 =cos(𝑧+𝑤)
• cos(𝑧+2𝜋)=ℯ^{𝑖(𝑧+2𝜋)}+ℯ^{−𝑖(𝑧+2𝜋)}2=cos𝑧
• sin(𝑧+2𝜋)=ℯ^{𝑖(𝑧+2𝜋)}−ℯ^{−𝑖(𝑧+2𝜋)}2𝑖=sin𝑧
• cos²𝑧+cos²𝑧=1. Proof: Let 𝑤=−𝑧 in the addition formula for cosine.
• sin(𝑧+𝜋2)=cos𝑧

  Proof:

  sin(𝑧+𝜋2)=ℯ𝑖(𝑧+𝜋2)−ℯ−𝑖(𝑧+𝜋2)2𝑖  =𝑖ℯ𝑖𝑧−(−𝑖)ℯ−𝑖𝑧2𝑖  =ℯ𝑖𝑧+ℯ−𝑖𝑧2=cos𝑧
• sin𝑧=0⇔𝑧=𝑘𝜋, 𝑘∈ℤ

  Proof

  sin𝑧=0⇔ℯ𝑖𝑧−ℯ−𝑖𝑧2𝑖=0  ⇔ℯ𝑖𝑧=ℯ−𝑖𝑧  ⇔𝑖𝑧−(−𝑖𝑧)=2𝑘𝜋𝑖, 𝑘∈ℤ, the periodicity of the exponential with period of 2𝑘𝜋𝑖.  ⇔2𝑖𝑧=2𝑘𝜋𝑖, 𝑘∈ℤ  ⇔𝑧=𝑘𝜋, 𝑘∈ℤ
• cos𝑧=0⇔𝑧=𝜋2+𝑘𝜋, 𝑘∈ℤ

  Proof

  cos𝑧=0⇔ℯ𝑖𝑧+ℯ−𝑖𝑧2=0  ⇔ℯ2𝑖𝑧+12=(ℯ𝑖𝑧+𝑖)(ℯ𝑖𝑧−𝑖)2=0  ⇔ℯ2𝑖𝑧+1=0  ⇔ℯ2𝑖𝑧=−1=ℯ𝜋𝑖  ⇔2𝑖𝑧−𝜋𝑖=2𝑘𝜋𝑖, 𝑘∈ℤ, the periodicity of the exponential with period of 2𝑘𝜋𝑖  ⇔2𝑖𝑧=(2𝑘+1)𝜋𝑖, 𝑘∈ℤ  ⇔𝑧=𝜋2+𝑘𝜋, 𝑘∈ℤ
• Derivative of Sine: 𝑑𝑑𝑧sin𝑧=cos𝑧

  Proof

  𝑑𝑑𝑧sin𝑧=𝑑𝑑𝑧ℯ𝑖𝑧−ℯ−𝑖𝑧2𝑖  =𝑖ℯ𝑖𝑧−(−𝑖)ℯ−𝑖𝑧2𝑖  =ℯ𝑖𝑧+ℯ−𝑖𝑧2=cos𝑧
• Derivative of Cosine: 𝑑𝑑𝑧cos𝑧=−sin𝑧

  Proof

  𝑑𝑑𝑧cos𝑧=𝑑𝑑𝑧ℯ𝑖𝑧+ℯ−𝑖𝑧2  =𝑖ℯ𝑖𝑧+(−𝑖)ℯ−𝑖𝑧2  =𝑖(ℯ𝑖𝑧−ℯ−𝑖𝑧)2=−sin𝑧
• Complex sine in terms of real functions, sin𝑧=sin𝑥cosh𝑦+𝑖cos𝑥sinh𝑦

  Proof

  sin𝑧=sin(𝑥+𝑖𝑦)  =sin𝑥cos(𝑖𝑦)+cos𝑥sin(𝑖𝑦)  =sin𝑥ℯ𝑖(𝑖𝑦)+ℯ−𝑖(𝑖𝑦)2+cos𝑥ℯ𝑖(𝑖𝑦)−ℯ−𝑖(𝑖𝑦)2𝑖  =sin𝑥ℯ−𝑦+ℯ𝑦2+cos𝑥ℯ−𝑦−ℯ𝑦2𝑖  =sin𝑥ℯ𝑦+ℯ−𝑦2+𝑖cos𝑥ℯ𝑦−ℯ−𝑦2  =sin𝑥cosh𝑦+𝑖cos𝑥sinh𝑦
• Complex cosine in terms of real functions, cos𝑧=cos𝑥cosh𝑦+𝑖sin𝑥sinh𝑦

  Proof

  cos𝑧=cos(𝑥+𝑖𝑦)  =cos𝑥cos(𝑖𝑦)−sin𝑥sin(𝑖𝑦)  =cos𝑥ℯ𝑖(𝑖𝑦)+ℯ−𝑖(𝑖𝑦)2+sin𝑥ℯ𝑖(𝑖𝑦)−ℯ−𝑖(𝑖𝑦)2𝑖  =cos𝑥ℯ−𝑦+ℯ𝑦2+sin𝑥ℯ−𝑦−ℯ𝑦2𝑖  =cos𝑥ℯ𝑦+ℯ−𝑦2+𝑖sin𝑥ℯ𝑦−ℯ−𝑦2  =cos𝑥cosh𝑦+𝑖sin𝑥sinh𝑦