例题 f'(z)=ux(z,0)-iuy(z,0)=ez+sinz+icosz=ez+ie-iz. 于是 fe-ro=rwaw=e2-e0i=e-ea, 而f(0)=u(0,0)+iv(0,0)=0,从而f(z)=e2-e-iz,其共轭调和函 v(x,y)=Im f(z)=exsiny+ey sinx. 例题 𝑓′ 𝑧 = 𝑢𝑥 𝑧, 0 − 𝑖𝑢𝑦 𝑧, 0 = 𝑒 𝑧 + sin 𝑧 + 𝑖 cos𝑧 = 𝑒 𝑧 + 𝑖𝑒 −𝑖𝑧． 于是 𝑓 𝑧 − 𝑓 0 = න 0 𝑧 𝑓′ 𝑤 d𝑤 = (𝑒 ቚ 𝑧 − 𝑒 −𝑖𝑧) 𝑤=0 𝑤=𝑧 = 𝑒 𝑧 − 𝑒 −𝑖𝑧 ， 而 𝑓 0 = 𝑢 0,0 + 𝑖𝑣 0,0 = 0，从而 𝑓 𝑧 = 𝑒 𝑧 − 𝑒 −𝑖𝑧，其共轭调和函 数 𝑣 𝑥, 𝑦 = Im 𝑓 𝑧 = 𝑒 𝑥 sin 𝑦 + 𝑒 𝑦 sin 𝑥．